Long time dynamics of random data NLS and invariant measures
Nahmod, Andrea
In this talk we show how certain well posedness results that are not available using only deterministic techniques (eg. Fourier and harmonic analysis) can be obtained when introducing randomization in the set of initial data and using powerful but still classical tools from probability. These ideas go back to seminal work by J. Bourgain on the almost sure global well posedness and invariance of Gibbs measures for NLS and other dispersive PDE. After explaining some of these ideas , we describe in more detail some further ideas in recent probabilistic well posedness results for NLS (joint with G. Staffilani) and new work on the existence and uniqueness 3 of non-equilibrium in variant measures associated to resonant NLS (joint with Z. Hani, J. Mattingly, L. Rey-Bellet and G. Staffilani)
 All right, so let me first thank very much Frank and Jeremy and Tomás for the invitation to speak at this conference, but also to the other organizers for organizing such a great program and it's great to be here. So what I'm going to talk about is kind of a little bit the plan is and I apologize to some of you who have heard me at least once and I can see Alex there. So like, Michael said, feel free to prove a lemma while I speak on your own work. So anyway, so what I would like to do is because of the mixed audiences to first maybe review a little bit some of the work of Burgen, which is in the background of all of this, and they sort of talk about some recent work with Gileola Stapilani for the cubic equation in 2D and the quintic focusing in 1D and then sort of like talk about some more recent work, which we're still writing with Sacherhane and Jonathan Matty and me look really at Gileola and maybe perhaps between 1 and 2 and 3, I want you to think about like, it's kind of like talking about sort of in the first two is like the notion of Gibbs measures, which is this sort of like equilibrium states come in and while the second part is about sort of like the non-equilibrium states. So also, okay, so let me quickly so I want to talk about the Schrodinger equation and just for the purpose of this talk, I'm going to talk most of the time about the periodic case and I'm going to talk about the square torus, not the rational one. And sort of for the purpose of this talk, every time that I talk about critical or supercritical or subcritical always mean relative to scaling. And so as everybody knows here, because we have people here who have done that work, there has been a lot of progress in studying these equations and most of it has been sort of in deterministic aspects of the phenomena. But there's still some questions that remain. There's essentially like, you know, supercritical case, which even in the defocusing case people don't know if you have blow up. But also there are gaps in the in the well-posed theory, as we will see. So here is just quickly some reviews like of some of the work that you all know on RD. And so this is for subcritical regimes here, the work of Bergen, the work of Colliano-Cristafalian and Taco-Catal. And then we know sort of for critical equations, I mean, and I'm not putting all names, but there has been a lot of like fantastic work and also some conditional results that sort of intercritical. And so and supercritical, as I said, there is not much, there's not nothing known really in the defocusing case. Okay, so on the periodic case, though, the situation is much more limited and the results are fewer. And there's even a problem at the local well-posed theory, right? So Bergen proved, like in 93, that actually dispersion in the periodic is indeed weaker. And so for example in 1D, the L6, so the L4 in 2D actually do have epsilon derivatives. This is actually, he constructed the examples. And so what that means is that in the local well-posed theory, you cannot, although this equation, for example, the cubic in 2D or the quintic in 1D are L2 critical, you cannot close them. There is no local theory at the level of L2. That's open. Okay? S is bigger than zero. There are no results in L2. Local. Okay? And so in the same paper, the subcritical regime, so Bergen sort of proved, as I said, the local well-posed. And these are just some examples. And of course, you have global at the energy level. Okay? So how about global? Well, global, you have the work for, so there were work for small data, for Tess and Cofftetaru. And then the sort of the breakthrough work, the first sort of like global, large data global of Jonas Komposadere, who actually did the large data for the quintic defocusing in 3D. A student of mine did, after the breakthrough, actually, of Bergen, the matter of the strikers, he did, that's part of his thesis, he did the cubic defocusing in a lesson 4D. And as I said, there was a small data result before. And maybe I should mention here that there's work of Kilipandisan, which they have also a different proof of the small data and new proofs for the irrational case. But as I said, I'm going to focus on the rational case. And the reason is because the non-deterministic setting, you still rely heavily on analytic number theory. And so we don't know still how to treat many of the things in the rational case. So the point is that, as I said in the L2 level, this is not even known. So the point of view that you want to take is that then maybe what you should do is you should study these equations from a non-deterministic point of view. And this brings us to Bergen, which is the starting point in 1994-96, where he studied long-time dynamics for periodic equations in the almost sure sense, and showed that you have actually global well-posedness on a set of data, a full Gibbs measure. So the Gibbs measures for NLS have been constructed, for example, for the cubic NLS in 1D, 2D, and 3D defocusing, have been constructed by Glimm-Jaffin, the context of quantum field theory, like the P4 model, or the Stochastic Allen-Kahn. And then in the context of Hamiltonian systems by Leuvelius-Rosenspier, who also treated the focusing case in one dimension. Okay, here you have, if you don't know, that just in variance means that you have this measure that if you take a set of initial data, as you flow it, it preserves the measure. And so why is it that it works? I mean, because usually what happens is, when you are sitting on the, and I'll show you in a minute, when you are sitting in the statistical ensemble, or in the support of the measure, you are at a regularity which is very rough, and there are no concert quantities at that level. So why is it that you can get global? Okay, so the point is that, assuming that you have a local well-posedness, and it doesn't have to be deterministic, it could be non-deterministic, then the invariance of the measure where it acts as a concert quantity that allows you to continue the results from local to global. At a level of regularity where there are no concert quantities, so that's the key. Okay? And the virtue of doing this is that somehow it captures generic behavior, and I'll say something more in a minute about that. And so now this, yes, I will explain it a little. So an idea of what do I mean? And so, I mean, generic in the sense of probability, that's what I mean. Okay? But I'll, you know, infinite dimensions is a big place. But, you know, all right. So actually there are some, of course, limitations and challenges. One is actually for this perceived PDE, the actual constructions of these measures. They are not that easy. They're easy to construct if you are on finite measure, on the domains, if you're in infinite measure or infinite volume, there are some problems. If the PDE is not Hamiltonian, it's not always clear how to do it. And as I said in 2D and 3D, in the focusing, you actually, you cannot construct it directly, it was proved that actually what you have to do is you have to renormalize the nonlinearity and remove some sort of like divergences of some infinities in order for the measure to make sense. That was known. In the focusing case for 2D, Bridges and Slate said that there was no Gibbs measure. The measure is not available in higher dimensions like 4. And even in 3D, if instead of cubic, you are quintic, then it's also known that if you try to do the same renormalization procedure, you either go to something trivial or to the Gaussian measure, so there are no Gibbs measures either. And for example, if you take something like the defocusing quintic in 2D or the cubic in 3D for which there is a measure, the problem, and I'm going to show you why in a minute, is that in order to prove the invariance, you of course have to have a global flow and you can have actually a weak solution. But if you want to show the global well postness, then you are going to see it as you go high in dimensions, you're going to see it in rougher and rougher spaces, and it's actually not clear how to do this. This is challenging. The defocusing quintic in 2D might be within reach thanks to the new techniques from some sort of like the baradiff... baracontrol distributions for guvinelli or regularity structures, but this is a little too soon to talk about that. So quickly, how is this measure defined? Because I'm going to be needing this, so of course you think of your equation as an infinite dimensional Hamiltonian system on the Fourier coefficients. And let's say one dimension, let's take just one dimension, p less than equal to n5, we'll also have Spear said, well, you can construct this measure that formally looks like this. So this is like Lebesgue measure, this is Hamiltonian, and this is the measure. Now if you look at this, this is total nonsense, because there are three terms here, the Lebesgue measure and the kinetic and the potential energy, and every single factor is infinity. So this makes no sense, but it's a justive of what is it that you want to do, so you cannot define even the Lebesgue measure. But what you do is you construct it using Gaussian measure as a reference measure, and then you construct the Gibbs measure as a weighted measure in two steps. So the Gaussian measure on some sort of like Heeler space, whatever you wish, you construct it as the weak limit of finite dimensional measure. So here you see this is the kinetic part of your Hamiltonian, and this is the Lebesgue measure, which if you're Hamiltonian, the Lebesgue theorem tells you there is volume preserving. And then this raw ending is sort of like you can view it as an induced probability measure under this map, okay? So where these are just from now on, these are always IID complex Gaussian random variables taken with mean zero. And so the point now, the most important point now is that you want to understand the Gaussian measure raw as a weak limit of these finite dimensional approximations, and it's easy to see that they are additive, but they are not countably additive. And so the point is that if you want them to be countably additive, and there is a theorem, a trace theorem that tells you when they are countably additive, then the point is that on HS, they are countably additive only if S is less than a half, okay? But not in Sb or equal to a half. They are in 2D, they are countably additive if you are below L2, with probability zero more. In 3, if you are below minus a half probability zero, if you are above, okay? So what that means is that if I want to, this is exactly what guides you, what is the support of the measure, and where is the data going to leave? That's the regularity of the data you're going to take. So in 1D, if you're looking at the Gaussian measure, you have to be sort of in H a half minus. If you are in 2D, you have to be just below L2 and so on. So you see as you go high in dimensions, you get very, very rough. And so once you do this, then you know that this Gaussian measure is just the law for this random variable. And so these are the typical elements in the support of the measure. This is how they look, okay? And so this is a function almost surely in 1D and in distribution, in higher D, and you have these tail estimates. So as I said before, once you have constructed the Gaussian measure, then the Gibbs measure is just a weighted measure where sort of this is the nonlinear part of the Hamiltonian, this is a renormalization. And what you want is this to be sort of the Radon-Nicholim derivative. You want this to be in L1 relative to rho. And so if you are in the focusing case, you have nothing for free because you still in order to be, to do the renormalization for this to be probability measure, you have to put the L2 cutoff, okay? So the same cutoff that you see in the deterministic case comes here. And actually it's a nice exercise to see which constant is larger. And actually this one is smaller. So you don't win anything. But anyway, so if the power is less than 5, you just need some cutoff. And if the power is 5, which is the critical, mass critical, then it has to be sufficiently small. All right, so what is that Morgan D? It's like Morgan proving 1D. This is his result. So I just, I'm just stating it because I want you to get used to how he reads, is that you take the equation with that data, which is the data in the support of the measure. And then what does it mean to prove all more sure? It means that you can find a set in here. If you are focusing, you have to condition everything to have a small L2 of measure 1 of the, of the gives measure 1, such that for any such data, the initial value problem can be continued globally, okay? And then you also prove that this measure is also invariant, okay? So why is this important? Well, let's recall for a second, deterministically, you have local, if this is strictly bigger than zero, and globally, if you are in H1. And, and that's it. And so what the invariance measure gives you, as I said before, is a global result in H1-2, where there is nothing, there's no conserved quantity. So how are you going to get it? But that's, that's what the, the measure does for you. And so how does he prove this? Let me just quickly go through this, because we are going to need it. It's like, what, what you have to do is you have to approximate the equation by a finite dimensional approximation. And, and, and actually the crucial fact, because what I want to bring home here is, is how do you use the invariance of the measure? And what is important, which is a little bit of a, of a, of a confusion is that what you, what you really use is the invariance of the finite dimensional measure, not the, not the invariance of the infinite. Variance of the infinite sometimes comes in parallel to the, to the global well-posedness at the end. So, so why is it that the finite dimensional measure is, is, is invariant? Well, because for this equation, when you take these things, you stay Hamiltonian. So this stays Hamiltonian, which is not always true when you take an equation and in order to prove a local result, you have to gauge. I mean, what happens is that you might stay Hamiltonian, but the problem is that you don't know how it looks. And this is actually a very delicate point. What happens with Gaussian measures or measures under sort of these non-canonical transformations. And then as I said before, you will see it and then tells you that because you stay Hamiltonian, this is volume preserving, this is conserved. So this is a conserved, this is, this measure is, is invariant. And now what you do is you use the invariance of the finite dimensional measure to continue. So you know, by the, by the local theory, you have a deterministic local theory for this local. And so now you use the invariance of this to continue the solution to the finite dimensional approximation to a global one in time, provided that you have, you are on a set of good data. So what that means is that in this process, you have to select a set of, of omegas with probability one that tell you that you're still, that you can continue, that you can do this process. So once you have this, then how do you continue? You, there is this approximation lemma that tells you that once you have the finite dimensional reach time t, you take the u that you know that is local and you walk the u in an iterative state, state, a method side by side with the u n. And this is not a trivial problem because what happens is that you see after the initial the local time of existence, you somehow this two can become really far apart. Okay. And you have no a priori bound on u. So this is not trivial lemma. So what you have to do is you have to take as a stepping stone, an intermediate system, which is the infinite dimensional equation with projected data. And then what you do is you, you, you use that as a stepping stone and you compare u to u prime and then you pride to u n step by step. And then that's, that's how you do it. And so once you have that, now you have now the, the sort of the global you and sort of the icing in the cake is that you get the, you prove that you have these converges to, to, to mu and you prove you get the invariance of mu. But it's really the invariance of, of, of mu n. Okay. So how about 2d, what did Morgan did in 2d and here is where it becomes interesting. Maybe what I want to say is that so in 2d he considered now the cubic, which is still a little too critical. But now, as I said before, the Gibbs measure, which was known to exist, is, needs some sort of like renormalization in order to exist. Okay. And you can actually see this renormalization also in trying to prove this theorem because it's actually the same thing you need to remove to deal with resonances. And so what you do is you have to treat what is called the weak order nonlinearity. So you have to subtract this term from the cubic. And then he proved exactly the same theorem that then your globally will pose below L2. And the associated measure is invariant. So here, now this is the point that in the, in the kinetic, in the potential energy, this is unbounded almost surely. So you have to sort of weak order and so you renormalize the finite dimensional approximation and, and that removes this term. And what I would like to say is that actually this is the first result ever that is global and supercritical. Okay. So in the, in the dispersive community, this is the first result that is supercritical and global. So is it critical in the sense of L2 even? Yeah, scaling. Every time, as I said, every time I talk about critical, supercritical is relative to scaling. Okay. And so, and so something nice that is actually, just an observation is that you see that the measure sort of always charges open sets with positive measure. So if you prove using this argument, if you can prove that such a result with, with probability, with measure one, then what that tells you is that you cannot have blow up, blow up in that topology that is, you can have blow up, but you cannot be stable in that topology. Okay. So that's actually something nice. Okay. Because, you know, okay, you can still have this is the focus and you don't know if you're going to blow up, but it cannot be stable. Okay. So that tells you that that's kind of like this. Okay. So now the additional difficulty here is that Burgen did not have a local well post in place as he did in 1D. Okay. So he cannot even start the argument that I just described a minute ago because there's no local, there is no deterministic local well post in this before below L2 is supercritical. So what he says is, he says, well, look, I mean, I just want to prove a normal sure global result. So why do I need a deterministic local? It's enough to prove that I have a local well postness for data that is in the support of the measure. In other words, all I need is a probabilistic local well postness result. And so if you look at this paper from 96, which is a very long paper, it's like 56 pages, like, you know, 50 of those are just to prove the probabilistic local well postness. Okay. So that's, that's, that's the, that's the heart of matter in that paper. And the rest just follows the same as before. So that's what I said there. Okay. And this is what I said. I mean, so this brings us to this notion of saying, okay, so now forget about for a moment about the use measure and forget a moment about global. Let's think about just the local well postness. And so what, what Burgen is saying essentially is that if you, if you, if you, if you take data and you randomize the data, then you're always going to be able to improve your local theory. Okay. So that's the takeaway. Okay. And this is actually not a foreign idea for, for, for us and for, for anybody, but, but especially for harmonic analysts, because this is exactly what, what, what, what, what we know and we use all the time. When you prove little pale inequalities, for example, that sort of random series enjoy better LP, LP estimates. And so this randomization doesn't improve the regularity. What it does is improves the LP, the, the integrability. But once you have that, then you use that in turn to get better estimates that you would otherwise deterministically. So what is his strategy to prove this? And this is actually the same strategy that I don't know if at the same time, but, but in the time, this is the same strategy but what is used, used by the Pratt and the Bush for the, for the stochastic Alencona for the FIFOR model. And they were using exactly the same idea. What's the idea? The idea is that now you look solutions of a particular form. So what you'd say is you say, well, you randomize the data now what I'm going to do is I'm going to look for solutions that look like the linear evolution of the random data plus some other W. And instead of solving for you, you solve for the difference equation. Okay. And so as a consequence of this, and so what he can prove is that if you solve for this difference equation, now you solve here in a smoother space that you start. Okay. So you solve for a fixed, you know, whatever you want to do, but you're going to find the W in a smoother regularity than the initial data. And so as a consequence, he proves almost surely in omega that the nonlinear part always is a smoother than the linear part. Okay. This is in this 96 and 94 or 96 paper. Now an important remark here is that by doing this, it's not that you're just saying that if you randomize, you bring the problem from being like super critical to being subcritical. That's not true. I mean, what it is is that you have a problem where the nonlinearity is sort of hybrid, right? I mean, you have supercritical terms which are random, and you have sort of like W's which you treat deterministically, which are really maybe smoother or subcritical. But you have to treat all interactions, random, random, deterministic, deterministic, random, deterministic. And so one of the things that you need in place before you even start this thing is because you have to deal with the deterministic, deterministic at a smoother space, you actually have to have in place a deterministic local well postness in a smoother space to start, at least in this approach. All right. So let me just quickly tell you why it works, so how it works. So everybody, I'm sure here just stated by everybody knows sort of the typical large deviation estimates. So think of k here as being how many random terms you are dealing with in your nonlinearity, whatever that is. And so here you have just a sort of a linear combination of the random variables. So a large deviation tells you that in omega, sort of the LP norm is controlled by the L2 norm. And so what's the point of this? The point of this is that then by Chevich's, the set where the absolute value of this function is bigger than lambda is the case exponentially. So now take any delta and take lambda here to be delta to the minus k over 2 times the L2 norm and put it here. And so what that tells you is that there exists a set, okay, which is the complement of this. So this omega is the complement of this. So the probability of the complement of omega is the probability of this, which if you put this number here gives you that. That tells you that if you're outside that set, right, so if you're outside this set, so that means if you're here, then you can replace the absolute value of this by the L2 norm, square. And that's fundamental. So let me show you why. Okay, that's the key. So what's why? So let's do one example. Suppose that you have data that looks like this, okay. This is what we're going to have if you are in two dimensions just below L2, it will look like this, okay. And in doing your random-random interactions, what you have to do is you have to estimate things of this form, okay. Where this is just the set of n1 and 2 and 3, each of them in Z2, such that they behave like this. Now what the wake ordering does for you, which is actually very interesting because it's a different way of understanding resonance, is that it tells you that n1 and n3, these two are never n2, which means that you never lose the independence of the random variables. So every time that n1 is n2, this becomes an absolute value square, you lose the independence. So the wake ordering, what it does, which is the same as the resonance, it tells you that this stays, you always have independence, okay. And so now if you just were to do Cauchy-Schwarz here, if you want to estimate the absolute value of this, sorry, the L2 norm of this, and you look at this, and you simply were to, so you sum in n and in n because I'm doing L2, and you simply do Cauchy-Schwarz, then you pick up, you lose because you pick up the cardinality of that set. You pick up the cardinality of this set, which is big, okay. It's big. These are integers. You're counting lattice points on spheres or intersections of spheres or whatever. And then you lose, okay. And what that loss means that you cannot lose derivatives, you cannot close. And so what the large deviation estimate tells you is that you are replacing, as I said a minute ago, modulo, a set of omega, you can replace the absolute value with the L2. And so what you are looking at is at this, without that. And that's enough to close, okay. So that's how it works. I mean, that's how it works in the old run, okay. Now, I want to say that something just because Sergio's here, he was just saying quickly, that actually, well, maybe I'll say it in a minute, okay. Let me actually move a couple of transpositions. So the takeaway from here is that the randomization, what it does, is actually allows you to improve the local well-pulsed theory. And you should view the randomization as a separate issue from the invariance of the measure. Once you have improved your local well-pulsedness, the question is how do you go to global? And then it depends which equation you have, which regime you are. If you have a measure, you don't have a measure, you can actually do a variety of things, or you may not be able to do anything, at least with current technologies. At least nothing that is conditional. I mean, one thing that I should say here, maybe going back, is that one problem that you have is that you see, even if you have conserved quantities, or even if you have conserved quantities for you, in the way that you study this problems, when you study the difference equations, you have no conserved quantities. I mean, you lose, the W doesn't satisfy anything. So it's a bit problematic. Okay, so as I said, then how do you pass, so separate the two issues. Randomization improves local, and then if you have a measure, then you can go from local to global. If you don't have a measure, maybe you can do energy estimates, or you can do something else, depend on the regime, or you are still, you may be still working on it, because you have nothing. And as I said, there was a, this has been a lot of work after Burgen. So nothing happened between Burgen's work, and sort of around 2007. And there was a lot of activity. I'm not being comprehensive here. There's a lot of people that have worked for a lot of equations, and not just dispersive, also sort of like on fluids equations. And maybe because it's just search juice here, I wasn't going to say anything, because it's kind of like, this is actually part of a long term project. But you know, in this paper that we sort of consider sort of an linear wave equation with the climate market, null forms, the QIJs. And actually, this, we are interested in this question for, in this equation for a geometric purpose, and for trying to understand these methods beyond the, this regime in which what you do is you look around the linear evolution. We want to look around special solutions, and this, and these geometric equations have like a sort of ground state. So we are interested in trying to understand how to sort of understand, how to propagate a little bit better, how to understand the transport of the randomization. And, but what I want to say is that, you know, it's actually, it's a beautiful recast of these null forms that actually you can, they're exactly, I mean, they're like a gift, like keep on giving, because they are exactly what you need when you have quadratic derivatives to preserve the independence that I showed you before. So I told you that if you have random, random interactions, you have to do something like the weak ordering to make sure that you, you, you keep independent the random variables. And then, and then, and what the null form does is exactly that renormalization in the probabilistic context. And so, so it's actually quite interesting. And so let me quickly talk about this, this work. So this is actually two things in which, two results which are still at the subcritical regime. But the reason why we are interested in this is because they are actually closing an important gap in the subcritical theory between the deterministic theory and the, and the almost sure global well-posed. And so one is in 2D, you take the cubic defocusing equation. And as I said a minute ago, so what we're going to prove was that if you are below L2, you have almost sure global. And for the look, for the global deterministic using the i method, the best result known, which I have a picture here, is above two thirds. Okay. And so we're going to prove it here. And if you are here, you have global. But as I will explain in a minute, if you have a global well-posed that is a probabilistic, you cannot use just the deterministic propagation of regularity methods to, to, to fill in this gap. And so we were interested in understanding how to get global in this range. And for the 1D, which is focusing, you have the same problem, which there is a still a gap between, so that the almost sure global is just below half. And then of course you have in the focusing you have above H1 and the defocusing you have above four ninths. But in the focusing, you have nothing and we also wanted to close that gap. And both of the results follow from a method that we like to call propagation, probabilistic propagation of regularity. So, so let me put this in here because this is this what I just, I just said. This is, so let me just talk about the 2D. Why is this not trivial? The result is not trivial because if I give you data, which is a smoother than the random data that is sitting in H minus epsilon, that data is with probability zero, I told you, with probability zero is in the support of a measure. So I cannot just take a smoother data, a larger decay and use Boolean. It doesn't, you have nothing. Okay? So, so, so, so you usually, when you prove these results, I mean this is what is it that you do. Having in dispersive PD, having more regularity never helps you. Usually what you do is you say, well, you need a concert quantity to prove something global. You prove it at the level in which you have a concert quantity and then you use propagation of regularity to get the smoother solutions. And here that doesn't work. Okay? And so what we do instead is we sort of, the key idea is that what we want to do is we want to decompose the data into, into, into a term that is close to the support of, and I'm going to be made up precisely, close to the support of the measure and a rougher, in the rougher topology, in the rougher topology that where we are. So, so for example, in 2D we want to do it for S bigger than zero. So the rougher topology will be the H minus epsilon and then a smoother reminder. And the question is how to do this so that things work. And then we want to do sort of like a, sort of a perturbation argument to conclude. Okay? And, and the argument that I'm going to present here is, is actually we do it in these two cases. But it's very general. So every time that you have proved a result, that is almost sure using a measure and, and there is a gap above you, you can use it. So, so every time that you have a measure, you can use this every single time. So let me just give you this example. So this is the statement of a theorem. Sorry for it. I have to state something. So u alpha just means it's just to remind you that the data that I'm taking has this form. Okay? So this alpha means that this is the data. So you see when alpha is zero, you are in H minus epsilon. And this is with probability zero in the support of the measure. So u alpha just means that that's the data and you solve the cubic equation. And I'm above l two. So I don't, even if you normalize or renormalize your above l two, so you can always put the, the weak ordering, which is finite now into the linear term. So, so I'm not going to worry about that. And so the theorem says that if you give me any large t and any alpha positive and some, some epsilon, then there exists a set of, so this is the Gaussian measure. So there's no Gibbs measure, but you always have a Gaussian measure associated to the one plus alpha regularity, which is that means that you're, you're forming this with the derivative to the, to the alpha square instead of, instead of the gradient, you put the gradient one plus alpha square and the support is here. And so there exists a set with, with measure one such that you can continue this to, to a global solution and it has this form. Okay. So that's, that's the, the global result. So how do we prove this? You take the data and it's two steps. So in the first step, all we want is to, to, to be able to show that this data gives rise to, to a global solution in some rougher topology. Okay. And then in the step two, we will, we will optimize the regularity. So the idea is that you take this data and you, you do, you decompose it into, you take some L that is going to be determined later and you break it into N less than L. And then you take a second part where you put is like, if you wish, you can think of this AN as a, as a high, as a high pass filter. Okay. So you let, you let just this, this high frequencies pass. And that's what I call AN. And you write this as HL plus psi one omega. That's this term. And so what you know is, you know, this is just the low frequencies then in the, in the H epsilon one, which is the regular, then this is uniformly bounded for all. Okay. That's no, no, no problem. And, and this AN else are, are this, as I said, this sort of like high pass filter and in size, they're always less or equal than one over L. And so now what you do is you use Bourguin's result in H minus epsilon to claim that there exists a set with probability one such that if you look at now the YL, this YL, leave this alone, take this YL. Okay. And you add to that this term. Okay. So this is a smoother. So the probability, the Gibbs measure sees this with probability zero. And this is in the support of the measure beta L is just one over L. So this is in the support of Bourguin's measure. And you can add anything smooth to that because this is not seen by the measure. So this is still in the support of the measure. And so this is one of the elements for which Bourguin can, can, can continue globally. Okay. So write that as your YL plus an outside two. And so now you know that this evolves by Bourguin's result. This evolves globally to some solution UL that has this form. And you have this bounce because Bourguin proved that this is always a smoother than you start. You start here, you end up here. So furthermore, actually in part of the argument, another one to get in the details, what you get there is this, in this iteration argument, this approximation and this iteration argument that I mentioned before, the, the, you get as a sort of byproduct of that you get, so these are just the XSB norms with epsilon and B equals to a half. So you also get this uniform bounce on, on these terms in these XSB norms on each iterative time step. Okay. Uniform in L. So that's part of this approximation and iteration argument of Bourguin that I showed you before. So now what you do is you use this, that this is global, the UL is global, to prove the existence and uniqueness of the UAlpha that you want. Okay. And, and the key, the reason that's the, that's the perturbation argument. And the reason why the perturbation argument works is because the, these norms are less than one over L. They are small. So these terms are small. They can be made small. Okay. And so what you do is instead of solving for UAlpha, you look again at a, at a different, as a suitable difference equation. So you call zeta prime L to be the linear evolution of what you want minus the solution. This is the, this is what you want to solve. So you want to solve for that. But instead of looking at that, you look at the difference between the Bourguin one and that one. That's the perturbation. That looks like this. This, this solves this problem. And this is kind of like the cubic term. And this is the cubic term. And the fact that these two things are small is what allows you to, to, to finish with a contraction and iteration. I mean, that these two things are small plus this uniform bound allows you to prove that indeed this exists. And then the solution, the z alpha that you're after is simply this z prime. And so you found, you found the solution as the linear evolution of, of that data plus the z alpha. And once, yes, yes, once, no, no, no, he's cheating. Yeah, yeah, he's cheating. No, no, no, no, no. Come on. All right. So, and once you have that, this step two is the, is the, is the, is the recovery of regularity. Is the recovery of regularity, which actually I'm going to skip since, since now I'm being pressed. So the second part is, is what I want to talk is, so all of this was about, so I don't have this console. So the first part is about not the cosmic censorship, whatever, is about the global repositions which you don't like. So, but that gives measures which kind of is kind of like the, the, the, the equilibrium ones. And so what I want to talk about quickly now is about this other result, which is actually the transfer of energy. This is related to the talk of Patrick, maybe the first day for the halfway, but also to some lectures that's a her, I mean, it's kind of the other side of what's a her was describing last week. And so this is kind of like the, the, the typical question of the transfer of energy of the out of equilibrium dynamics for NLS. And so everybody knows that if you are on RDU and if you have scattering, there is no sort of like a skates and that on compact domains you lose that. Okay. So, so the question was how to, how to, how to capture that. And this is what also goes by the name of turbulence. And Burgen's approach was that what you want to do is you want to study the growth of the, of the sovereign norms to see how this moves, how this is transferred. Okay. So he's, he's, he's actually his conjecture in 2000 was whether he asked whether there exists global solutions to the cubic equations whose Hs norms growing definitely. Okay. This is what is called the sort of infinite cascade conjecture. And so Burgen himself did some progress on that. He constructed some special one D non-linear wave equations, some modified equations in which he, he exhibited that there's some work of Kuskin. And then the fundamental progress came in this paper of Koleander, Tils, Tafilani, Takaoka Tao in which they actually constructed large but finite growth on the sovereign norms. And if you haven't seen that result, I put it here. There's also work of Honey, Gerard, and Prelier, Guarde Kaloshin, Prostitius. And, and more recently, actually in a really nice paper of Honey Posaders at Govind, Vigilia, sort of they prove that if instead of the torus you take the product domain, these cylinders, then actually they, they, they, they solve the conjecture. They said they prove yes and they also gave some, some rate. And on Monday, the first talk of the, of this conference, we saw Patrick talking about similar things for the, for the halfway on R. And so now to understand Burgen's conjecture for the cubic analysis, actually very, very hard, very hard, of course, very hard problem, very ambitious. And so, but something that which is kind of like in between Burgen and what I described in the first part of the talk, which is this, this, this invariant Gibbs measures is really the, the, the study of the existence and uniqueness of non-equilibrium invariant measures. Okay. So this is kind of like the, the, in the stack mech, is kind of like this is very hard and very poorly understood the, the, the existence of stationary non-equilibrium states. And, and for us this, this is trying to maybe give some sort of justification, some sort of like rigorous sort of framework for, for the statistical description of the sort of auto-equilibrium dynamics. Now this problem is very hard and it's very hard even for stochastic-califorced systems in which is actually, is actually very hard, but there has been progress in this sort of like chains of oscillators of Eichmann, Piller, Reivelle, Reivelle, Thomas and more recently of Hermatingly in which they look at sort of a collection of unharmonic oscillators with, with nearest, near-neighborhood coupling. So here you have the Hamiltonians they consider and this is in the first two papers, this is the relationship between the pinning and the coupling and this is the much harder case in which the relationship is like this and they're already, they cannot solve this problem unless they only have three oscillators. So here there is no, no, no sort of problem with the end but here they cannot do more than three. And the idea is that what you do is you take this and you put them into contact with two heat baths at different temperatures. So you have your system and what you do is you want to inject energy into the first mode and dissipate it into the last and you want to try to see whether the system will relax into, into a stationary non-equilibrium state. And so for us, so, so for us the starting point is actually this toy model in the work of Colliander-Klisthafilani Tagocca Tao whose Hamiltonian has this form and, and it is known that proved there. So here this is much harder than the other one because the interactions don't depend just on the relative distances but also on the momenta of the particle and the neighbors. And there's known that there is a finite set for which the resonant NLS is close and collapses to, to this toy model, the one that has this Hamiltonian. And so we do the same thing, we attach two heat baths at the low mode C1 and Cn and we just sort of a standard way of, of adding and dissipating energy. And what we expect is this system to, to converge to a non-equilibrium invariant measure which has some sort of flux from low frequencies to hard and we also want some sort of like rate of convergence. And so that's, that's what we want to do. And so the stochastic of the model looks like this for that H. So this is, you see the heat baths here and we also add these terms, that's the Hamiltonian. So we leave the middle terms, the middle modes alone and we just do this. And so you look at the, so just, I wrote it just for convenience, this is the, the sort of the, the generator of the, of the transition semi-group, the, the Fokker Planck operator is here. And so what you do with the first thing that you can prove is that if you are at equal temperatures of T1 is Tn, then this system indeed relaxes to an invariant Gibbs measure. Okay. And all, all of that follows from the fact that if you look at the adjoint of the, of the operator, Fokker Planck on this, this is zero. And so what is interesting is in the non-equilibrium case, that's a sort of equilibrium, is what happened at different temperatures. Okay. And you want to prove whether there exists a unique smoother-guiding non-equilibrium invariant measures that has this transfer of energy. And the answer is that again, that if you have three modes, so in other words, we only have one, two, three, so J is two. Then actually we can prove that and maybe to finish because I don't want to push your patience. What does it have? Thirty seconds? No, no, it still has three minutes. Oh, three minutes, perfect. Wonderful. So is that, what I want to explain is that the proof of this, what is hard is, is the only thing that is really hard is the existence. Okay. Everything else kind of is a standard. Okay. But the existence of the measure is very, very hard. So to understand this, let me show you two transparencies. First, first I'm going to maybe show you what we can prove. So we can also get some rates. And the problem is that we're still finishing. There is one region which is a little delicate, so we don't know if we are going to get exponential or polynomial yet. So one useful thing is to change coordinates. Then you have to change coordinates more than once, but you can change coordinates. So this is the ij, so this is m. It's the sum of these squares. And this is the angles. And so the first problem we have, unlike the previous works, is that we lose the hyperliphticity in all of phase space. So we are not hyperliptic everywhere, but we have to remove this. Okay. We have to remove, we have a degenerate line that we have to remove. The good thing is that if you start with c2 equals to zero, then you will remain at c2 equals to zero. But in the hyperliphticity, which is what gives you that if something exists is going to be smooth, then we have now a boundary. We have this boundary. And so the existence of the measure follows from what is it that you want to do. What you want to do is you want to construct the elliponopfunction that somehow penalizes this region where this is a small and sort of high frequencies. And somehow what you want to do is once you construct it, then that construction will give you an upper bound on the heating time of the good region, which should be a compact set. So let me show you quickly a picture, because maybe then it will be easy. This is actually what is how about the problem. We have to chop phase space, which they didn't have to do before. And so now it's like this is the good region where you want all your dynamics to happen. And so what you want to do is you want to prove, you want to construct the elliponopfunction here that tells you that as soon as you enter any of these regions, you are kicked back into here. Okay. But usually, you know, usually, when these things are very hard to do, and usually people what they do is they say, well, you know, your guess, your best guess is that the elliponopfunction should be e to the m. But that doesn't work everywhere here. And you have to actually understand very well what are the dynamics of the deterministic system to understand what works where. And in the regions in which you have more than one boundary, these boundaries are determined by the temperature and the gamma and so on. But in the regions in which you have these boundaries, they have contact, they talk to each other, you can guess one and then the way to solve it, the way to solve it. So you can guess one of them. And then what you have to do is you have to solve sort of a Poisson equation with the right boundary values and some convexity conditions and solve for all the other ones, flow it all the other ones to get them. And in some of the regions to get these, the problem is that you have to really understand to get this to really have to understand how the phases, you need to understand the phase diagram and know that the phases will look at some point. And so it's actually quite delicate. And so this is where the heart of the matter is. And as I said, once you have that, then the rest, the uniqueness and the ergo-dicity follows from sort of a control argument, a controllability dilemma that tells you that the deterministic system sort of can access any region in phase space. So if you give me two points, I can find an epsilon, a neighborhood of one. So that I start here, I end up in the other one, anywhere in phase space. And then there is a standard theorem of Strucambara that tells you that once you have that for the deterministic system, the stochastic one will behave the same way. And thank you very much. I'll leave you the picture. I spent a lot of time on that. Thank you. Questions? Yes. So, did I understand correctly that the difficulty is to understand the dynamics of this ODE in big dimensions in a very precise way? Yes, and we cannot do, you see, the problem is look at this. We have to break all of this only with three modes. If you have four, for example, what is the problem? The problem is, okay, higher energies, you can do, but the problem is at low energies, like when you're zero. So if you are stuck in the middle, your energy doesn't move anywhere. I mean, if you have four modes, maybe you are stuck there. And so, you know, we are happy first if we can do three. I mean, the paper of Jonathan with Martin, they're also on three. They cannot do four either. So it's not clear. It will be much more complicated. The way in which you will have to chop and understand is the construction of this Lyapunov function is hard. But does it help if you have more conserved quantities? I don't know. You see, the Hamiltonian is here. I don't know. Good question.
 All right, so let me first thank very much Frank and Jeremy and Tomás for the invitation  to speak at this conference, but also to the other organizers for organizing such a great  program and it's great to be here. So what I'm going to talk about is kind of a little  bit the plan is and I apologize to some of you who have heard me at least once and I  can see Alex there. So like, Michael said, feel free to prove a lemma while I speak  on your own work. So anyway, so what I would like to do is because of the mixed audiences  to first maybe review a little bit some of the work of Burgen, which is in the background  of all of this, and they sort of talk about some recent work with Gileola Stapilani for  the cubic equation in 2D and the quintic focusing in 1D and then sort of like talk about some  more recent work, which we're still writing with Sacherhane and Jonathan Matty and me  look really at Gileola and maybe perhaps between 1 and 2 and 3, I want you to think about like,  it's kind of like talking about sort of in the first two is like the notion of Gibbs  measures, which is this sort of like equilibrium states come in and while the second part is  about sort of like the non-equilibrium states. So also, okay, so let me quickly so I want  to talk about the Schrodinger equation and just for the purpose of this talk, I'm going  to talk most of the time about the periodic case and I'm going to talk about the square  torus, not the rational one. And sort of for the purpose of this talk, every time that  I talk about critical or supercritical or subcritical always mean relative to scaling.  And so as everybody knows here, because we have people here who have done that work,  there has been a lot of progress in studying these equations and most of it has been sort  of in deterministic aspects of the phenomena. But there's still some questions that remain.  There's essentially like, you know, supercritical case, which even in the defocusing case people  don't know if you have blow up. But also there are gaps in the in the well-posed theory,  as we will see. So here is just quickly some reviews like of some of the work that you all  know on RD. And so this is for subcritical regimes here, the work of Bergen, the work  of Colliano-Cristafalian and Taco-Catal. And then we know sort of for critical equations,  I mean, and I'm not putting all names, but there has been a lot of like fantastic work  and also some conditional results that sort of intercritical. And so and supercritical,  as I said, there is not much, there's not nothing known really in the defocusing case.  Okay, so on the periodic case, though, the situation is much more limited and the results  are fewer. And there's even a problem at the local well-posed theory, right? So Bergen  proved, like in 93, that actually dispersion in the periodic is indeed weaker. And so for  example in 1D, the L6, so the L4 in 2D actually do have epsilon derivatives. This is actually,  he constructed the examples. And so what that means is that in the local well-posed theory,  you cannot, although this equation, for example, the cubic in 2D or the quintic in 1D are  L2 critical, you cannot close them. There is no local theory at the level of L2. That's  open. Okay? S is bigger than zero. There are no results in L2. Local. Okay? And so in the  same paper, the subcritical regime, so Bergen sort of proved, as I said, the local well-posed.  And these are just some examples. And of course, you have global at the energy level. Okay?  So how about global? Well, global, you have the work for, so there were work for small  data, for Tess and Cofftetaru. And then the sort of the breakthrough work, the first sort  of like global, large data global of Jonas Komposadere, who actually did the large data  for the quintic defocusing in 3D. A student of mine did, after the breakthrough, actually,  of Bergen, the matter of the strikers, he did, that's part of his thesis, he did the  cubic defocusing in a lesson 4D. And as I said, there was a small data result before.  And maybe I should mention here that there's work of Kilipandisan, which they have also  a different proof of the small data and new proofs for the irrational case. But as I said,  I'm going to focus on the rational case. And the reason is because the non-deterministic  setting, you still rely heavily on analytic number theory. And so we don't know still  how to treat many of the things in the rational case.  So the point is that, as I said in the L2 level, this is not even known. So the point  of view that you want to take is that then maybe what you should do is you should study  these equations from a non-deterministic point of view. And this brings us to Bergen, which  is the starting point in 1994-96, where he studied long-time dynamics for periodic equations  in the almost sure sense, and showed that you have actually global well-posedness on  a set of data, a full Gibbs measure. So the Gibbs measures for NLS have been constructed,  for example, for the cubic NLS in 1D, 2D, and 3D defocusing, have been constructed by  Glimm-Jaffin, the context of quantum field theory, like the P4 model, or the Stochastic  Allen-Kahn. And then in the context of Hamiltonian systems by Leuvelius-Rosenspier, who also treated  the focusing case in one dimension. Okay, here you have, if you don't know, that just  in variance means that you have this measure that if you take a set of initial data, as  you flow it, it preserves the measure. And so why is it that it works? I mean, because  usually what happens is, when you are sitting on the, and I'll show you in a minute, when  you are sitting in the statistical ensemble, or in the support of the measure, you are  at a regularity which is very rough, and there are no concert quantities at that level. So  why is it that you can get global? Okay, so the point is that, assuming that you have  a local well-posedness, and it doesn't have to be deterministic, it could be non-deterministic,  then the invariance of the measure where it acts as a concert quantity that allows you  to continue the results from local to global. At a level of regularity where there are no  concert quantities, so that's the key. Okay? And the virtue of doing this is that somehow  it captures generic behavior, and I'll say something more in a minute about that. And  so now this, yes, I will explain it a little. So an idea of what do I mean? And so, I mean,  generic in the sense of probability, that's what I mean. Okay? But I'll, you know, infinite  dimensions is a big place. But, you know, all right. So actually there are some, of  course, limitations and challenges. One is actually for this perceived PDE, the actual  constructions of these measures. They are not that easy. They're easy to construct if  you are on finite measure, on the domains, if you're in infinite measure or infinite  volume, there are some problems. If the PDE is not Hamiltonian, it's not always clear  how to do it. And as I said in 2D and 3D, in the focusing, you actually, you cannot construct  it directly, it was proved that actually what you have to do is you have to renormalize  the nonlinearity and remove some sort of like divergences of some infinities in order for  the measure to make sense. That was known. In the focusing case for 2D, Bridges and Slate  said that there was no Gibbs measure. The measure is not available in higher dimensions  like 4. And even in 3D, if instead of cubic, you are quintic, then it's also known that  if you try to do the same renormalization procedure, you either go to something trivial  or to the Gaussian measure, so there are no Gibbs measures either.  And for example, if you take something like the defocusing quintic in 2D or the cubic  in 3D for which there is a measure, the problem, and I'm going to show you why in a minute,  is that in order to prove the invariance, you of course have to have a global flow and  you can have actually a weak solution. But if you want to show the global well postness,  then you are going to see it as you go high in dimensions, you're going to see it in rougher  and rougher spaces, and it's actually not clear how to do this. This is challenging.  The defocusing quintic in 2D might be within reach thanks to the new techniques from some  sort of like the baradiff... baracontrol distributions for guvinelli or regularity structures,  but this is a little too soon to talk about that.  So quickly, how is this measure defined? Because I'm going to be needing this, so of  course you think of your equation as an infinite dimensional Hamiltonian system on the Fourier  coefficients. And let's say one dimension, let's take just one dimension, p less than  equal to n5, we'll also have Spear said, well, you can construct this measure that formally  looks like this. So this is like Lebesgue measure, this is Hamiltonian, and this is the  measure. Now if you look at this, this is total nonsense, because there are three terms  here, the Lebesgue measure and the kinetic and the potential energy, and every single  factor is infinity. So this makes no sense, but it's a justive of what is it that you  want to do, so you cannot define even the Lebesgue measure. But what you do is you construct  it using Gaussian measure as a reference measure, and then you construct the Gibbs  measure as a weighted measure in two steps. So the Gaussian measure on some sort of like  Heeler space, whatever you wish, you construct it as the weak limit of finite dimensional  measure. So here you see this is the kinetic part of your Hamiltonian, and this is the  Lebesgue measure, which if you're Hamiltonian, the Lebesgue theorem tells you there is volume  preserving. And then this raw ending is sort of like you can view it as an induced probability  measure under this map, okay? So where these are just from now on, these are always IID  complex Gaussian random variables taken with mean zero. And so the point now, the most  important point now is that you want to understand the Gaussian measure raw as a weak limit of  these finite dimensional approximations, and it's easy to see that they are additive,  but they are not countably additive. And so the point is that if you want them to be countably  additive, and there is a theorem, a trace theorem that tells you when they are countably  additive, then the point is that on HS, they are countably additive only if S is less than  a half, okay? But not in Sb or equal to a half. They are in 2D, they are countably additive  if you are below L2, with probability zero more. In 3, if you are below minus a half  probability zero, if you are above, okay? So what that means is that if I want to, this  is exactly what guides you, what is the support of the measure, and where is the data going  to leave? That's the regularity of the data you're going to take. So in 1D, if you're  looking at the Gaussian measure, you have to be sort of in H a half minus. If you are  in 2D, you have to be just below L2 and so on. So you see as you go high in dimensions,  you get very, very rough. And so once you do this, then you know that this Gaussian  measure is just the law for this random variable. And so these are the typical elements in the  support of the measure. This is how they look, okay? And so this is a function almost surely  in 1D and in distribution, in higher D, and you have these tail estimates. So as I said  before, once you have constructed the Gaussian measure, then the Gibbs measure is just a  weighted measure where sort of this is the nonlinear part of the Hamiltonian, this is  a renormalization. And what you want is this to be sort of the Radon-Nicholim derivative.  You want this to be in L1 relative to rho. And so if you are in the focusing case, you  have nothing for free because you still in order to be, to do the renormalization for  this to be probability measure, you have to put the L2 cutoff, okay? So the same cutoff  that you see in the deterministic case comes here. And actually it's a nice exercise to  see which constant is larger. And actually this one is smaller. So you don't win anything.  But anyway, so if the power is less than 5, you just need some cutoff. And if the power  is 5, which is the critical, mass critical, then it has to be sufficiently small.  All right, so what is that Morgan D? It's like Morgan proving 1D. This is his result.  So I just, I'm just stating it because I want you to get used to how he reads, is that  you take the equation with that data, which is the data in the support of the measure.  And then what does it mean to prove all more sure? It means that you can find a set in  here. If you are focusing, you have to condition everything to have a small L2 of measure 1  of the, of the gives measure 1, such that for any such data, the initial value problem  can be continued globally, okay? And then you also prove that this measure is also invariant,  okay? So why is this important? Well, let's recall for a second, deterministically, you  have local, if this is strictly bigger than zero, and globally, if you are in H1. And,  and that's it. And so what the invariance measure gives you, as I said before, is a  global result in H1-2, where there is nothing, there's no conserved quantity. So how are  you going to get it? But that's, that's what the, the measure does for you. And so how  does he prove this? Let me just quickly go through this, because we are going to need  it. It's like, what, what you have to do is you have to approximate the equation by a  finite dimensional approximation. And, and, and actually the crucial fact, because what  I want to bring home here is, is how do you use the invariance of the measure? And what  is important, which is a little bit of a, of a, of a confusion is that what you, what  you really use is the invariance of the finite dimensional measure, not the, not the invariance  of the infinite. Variance of the infinite sometimes comes in parallel to the, to the  global well-posedness at the end. So, so why is it that the finite dimensional measure  is, is, is invariant? Well, because for this equation, when you take these things, you stay  Hamiltonian. So this stays Hamiltonian, which is not always true when you take an equation  and in order to prove a local result, you have to gauge. I mean, what happens is that  you might stay Hamiltonian, but the problem is that you don't know how it looks. And this  is actually a very delicate point. What happens with Gaussian measures or measures under sort  of these non-canonical transformations. And then as I said before, you will see it and  then tells you that because you stay Hamiltonian, this is volume preserving, this is conserved.  So this is a conserved, this is, this measure is, is invariant. And now what you do is you  use the invariance of the finite dimensional measure to continue. So you know, by the,  by the local theory, you have a deterministic local theory for this local. And so now you  use the invariance of this to continue the solution to the finite dimensional approximation  to a global one in time, provided that you have, you are on a set of good data. So what  that means is that in this process, you have to select a set of, of omegas with probability  one that tell you that you're still, that you can continue, that you can do this process.  So once you have this, then how do you continue? You, there is this approximation lemma that  tells you that once you have the finite dimensional reach time t, you take the u that you know  that is local and you walk the u in an iterative state, state, a method side by side with the  u n. And this is not a trivial problem because what happens is that you see after the initial  the local time of existence, you somehow this two can become really far apart. Okay. And  you have no a priori bound on u. So this is not trivial lemma. So what you have to do  is you have to take as a stepping stone, an intermediate system, which is the infinite  dimensional equation with projected data. And then what you do is you, you, you use that  as a stepping stone and you compare u to u prime and then you pride to u n step by step.  And then that's, that's how you do it. And so once you have that, now you have now the,  the sort of the global you and sort of the icing in the cake is that you get the, you  prove that you have these converges to, to, to mu and you prove you get the invariance  of mu. But it's really the invariance of, of, of mu n. Okay. So how about 2d, what did  Morgan did in 2d and here is where it becomes interesting. Maybe what I want to say is that  so in 2d he considered now the cubic, which is still a little too critical. But now, as  I said before, the Gibbs measure, which was known to exist, is, needs some sort of like  renormalization in order to exist. Okay. And you can actually see this renormalization  also in trying to prove this theorem because it's actually the same thing you need to remove  to deal with resonances. And so what you do is you have to treat what is called the weak  order nonlinearity. So you have to subtract this term from the cubic. And then he proved  exactly the same theorem that then your globally will pose below L2. And the associated measure  is invariant. So here, now this is the point that in the, in the kinetic, in the potential  energy, this is unbounded almost surely. So you have to sort of weak order and so you  renormalize the finite dimensional approximation and, and that removes this term. And what  I would like to say is that actually this is the first result ever that is global and  supercritical. Okay. So in the, in the dispersive community, this is the first result that is  supercritical and global.  So is it critical in the sense of L2 even?  Yeah, scaling. Every time, as I said, every time I talk about critical, supercritical  is relative to scaling. Okay. And so, and so something nice that is actually, just an  observation is that you see that the measure sort of always charges open sets with positive  measure. So if you prove using this argument, if you can prove that such a result with,  with probability, with measure one, then what that tells you is that you cannot have blow  up, blow up in that topology that is, you can have blow up, but you cannot be stable in  that topology. Okay. So that's actually something nice. Okay. Because, you know, okay, you can  still have this is the focus and you don't know if you're going to blow up, but it cannot  be stable. Okay. So that tells you that that's kind of like this. Okay. So now the additional  difficulty here is that Burgen did not have a local well post in place as he did in 1D.  Okay. So he cannot even start the argument that I just described a minute ago because  there's no local, there is no deterministic local well post in this before below L2 is  supercritical.  So what he says is, he says, well, look, I mean, I just want to prove a normal sure global  result. So why do I need a deterministic local? It's enough to prove that I have a local  well postness for data that is in the support of the measure. In other words, all I need  is a probabilistic local well postness result. And so if you look at this paper from 96,  which is a very long paper, it's like 56 pages, like, you know, 50 of those are just to prove  the probabilistic local well postness. Okay. So that's, that's, that's the, that's the  heart of matter in that paper. And the rest just follows the same as before. So that's  what I said there. Okay. And this is what I said. I mean, so this brings us to this notion  of saying, okay, so now forget about for a moment about the use measure and forget a  moment about global. Let's think about just the local well postness. And so what, what  Burgen is saying essentially is that if you, if you, if you, if you take data and you randomize  the data, then you're always going to be able to improve your local theory. Okay. So that's  the takeaway. Okay. And this is actually not a foreign idea for, for, for us and for, for  anybody, but, but especially for harmonic analysts, because this is exactly what, what, what, what,  what we know and we use all the time. When you prove little pale inequalities, for example,  that sort of random series enjoy better LP, LP estimates. And so this randomization doesn't  improve the regularity. What it does is improves the LP, the, the integrability. But once you  have that, then you use that in turn to get better estimates that you would otherwise  deterministically. So what is his strategy to prove this? And this is actually the same  strategy that I don't know if at the same time, but, but in the time, this is the same strategy  but what is used, used by the Pratt and the Bush for the, for the stochastic Alencona for  the FIFOR model. And they were using exactly the same idea. What's the idea? The idea is that now  you look solutions of a particular form. So what you'd say is you say, well, you randomize the  data now what I'm going to do is I'm going to look for solutions that look like the linear  evolution of the random data plus some other W. And instead of solving for you, you solve for  the difference equation. Okay. And so as a consequence of this, and so what he can prove is  that if you solve for this difference equation, now you solve here in a smoother space that you  start. Okay. So you solve for a fixed, you know, whatever you want to do, but you're going to find  the W in a smoother regularity than the initial data. And so as a consequence, he proves almost  surely in omega that the nonlinear part always is a smoother than the linear part. Okay. This is  in this 96 and 94 or 96 paper. Now an important remark here is that by doing this, it's not that  you're just saying that if you randomize, you bring the problem from being like super critical to  being subcritical. That's not true. I mean, what it is is that you have a problem where the  nonlinearity is sort of hybrid, right? I mean, you have supercritical terms which are random,  and you have sort of like W's which you treat deterministically, which are really maybe smoother  or subcritical. But you have to treat all interactions, random, random, deterministic,  deterministic, random, deterministic. And so one of the things that you need in place before you  even start this thing is because you have to deal with the deterministic, deterministic at a  smoother space, you actually have to have in place a deterministic local well postness in a  smoother space to start, at least in this approach. All right. So let me just quickly tell you why  it works, so how it works. So everybody, I'm sure here just stated by everybody knows sort of the  typical large deviation estimates. So think of k here as being how many random terms you are  dealing with in your nonlinearity, whatever that is. And so here you have just a sort of a linear  combination of the random variables. So a large deviation tells you that in omega, sort of the  LP norm is controlled by the L2 norm. And so what's the point of this? The point of this is that then  by Chevich's, the set where the absolute value of this function is bigger than lambda is the case  exponentially. So now take any delta and take lambda here to be delta to the minus k over 2  times the L2 norm and put it here. And so what that tells you is that there exists a set,  okay, which is the complement of this. So this omega is the complement of this. So the probability  of the complement of omega is the probability of this, which if you put this number here gives  you that. That tells you that if you're outside that set, right, so if you're outside this set,  so that means if you're here, then you can replace the absolute value of this by the L2 norm,  square. And that's fundamental. So let me show you why. Okay, that's the key. So what's why? So let's  do one example. Suppose that you have data that looks like this, okay. This is what we're going  to have if you are in two dimensions just below L2, it will look like this, okay. And in doing your  random-random interactions, what you have to do is you have to estimate things of this form,  okay. Where this is just the set of n1 and 2 and 3, each of them in Z2, such that they behave  like this. Now what the wake ordering does for you, which is actually very interesting because  it's a different way of understanding resonance, is that it tells you that n1 and n3, these two are  never n2, which means that you never lose the independence of the random variables. So every  time that n1 is n2, this becomes an absolute value square, you lose the independence. So the  wake ordering, what it does, which is the same as the resonance, it tells you that this stays,  you always have independence, okay. And so now if you just were to do Cauchy-Schwarz here, if you  want to estimate the absolute value of this, sorry, the L2 norm of this, and you look at this,  and you simply were to, so you sum in n and in n because I'm doing L2, and you simply do Cauchy-Schwarz,  then you pick up, you lose because you pick up the cardinality of that set. You pick up the  cardinality of this set, which is big, okay. It's big. These are integers. You're counting lattice  points on spheres or intersections of spheres or whatever. And then you lose, okay. And what that  loss means that you cannot lose derivatives, you cannot close. And so what the large deviation  estimate tells you is that you are replacing, as I said a minute ago, modulo, a set of omega,  you can replace the absolute value with the L2. And so what you are looking at is at this, without  that. And that's enough to close, okay. So that's how it works. I mean, that's how it works in the  old run, okay. Now, I want to say that something just because Sergio's here, he was just saying quickly,  that actually, well, maybe I'll say it in a minute, okay. Let me actually move a couple of  transpositions. So the takeaway from here is that the randomization, what it does,  is actually allows you to improve the local well-pulsed theory. And you should view the  randomization as a separate issue from the invariance of the measure. Once you have improved  your local well-pulsedness, the question is how do you go to global? And then it depends which  equation you have, which regime you are. If you have a measure, you don't have a measure,  you can actually do a variety of things, or you may not be able to do anything, at least with  current technologies. At least nothing that is conditional. I mean, one thing that I should  say here, maybe going back, is that one problem that you have is that you see, even if you have  conserved quantities, or even if you have conserved quantities for you, in the way that you study this  problems, when you study the difference equations, you have no conserved quantities. I mean, you lose,  the W doesn't satisfy anything. So it's a bit problematic.  Okay, so as I said, then how do you pass, so separate the two issues. Randomization improves  local, and then if you have a measure, then you can go from local to global. If you don't have a  measure, maybe you can do energy estimates, or you can do something else, depend on the regime,  or you are still, you may be still working on it, because you have nothing. And as I said,  there was a, this has been a lot of work after Burgen. So nothing happened between Burgen's work,  and sort of around 2007. And there was a lot of activity. I'm not being comprehensive here. There's  a lot of people that have worked for a lot of equations, and not just dispersive, also sort of  like on fluids equations. And maybe because it's just search juice here, I wasn't going to say  anything, because it's kind of like, this is actually part of a long term project. But you know,  in this paper that we sort of consider sort of an linear wave equation with the climate  market, null forms, the QIJs. And actually, this, we are interested in this question for,  in this equation for a geometric purpose, and for trying to understand these methods beyond  the, this regime in which what you do is you look around the linear evolution. We want to look around  special solutions, and this, and these geometric equations have like a sort of ground state. So  we are interested in trying to understand how to sort of understand, how to propagate a little  bit better, how to understand the transport of the randomization. And, but what I want to say is that,  you know, it's actually, it's a beautiful recast of these null forms that actually you can, they're  exactly, I mean, they're like a gift, like keep on giving, because they are exactly what you need  when you have quadratic derivatives to preserve the independence that I showed you before. So I  told you that if you have random, random interactions, you have to do something like the weak ordering  to make sure that you, you, you keep independent the random variables. And then, and then, and  what the null form does is exactly that renormalization in the probabilistic context. And so, so it's  actually quite interesting. And so let me quickly talk about this, this work. So this is actually  two things in which, two results which are still at the subcritical regime. But the reason why we  are interested in this is because they are actually closing an important gap in the subcritical  theory between the deterministic theory and the, and the almost sure global well-posed. And so one  is in 2D, you take the cubic defocusing equation. And as I said a minute ago, so what we're going  to prove was that if you are below L2, you have almost sure global. And for the look, for the  global deterministic using the i method, the best result known, which I have a picture here, is above  two thirds. Okay. And so we're going to prove it here. And if you are here, you have global. But as  I will explain in a minute, if you have a global well-posed that is a probabilistic, you cannot use  just the deterministic propagation of regularity methods to, to, to fill in this gap. And so we  were interested in understanding how to get global in this range. And for the 1D, which is focusing,  you have the same problem, which there is a still a gap between, so that the almost sure global is  just below half. And then of course you have in the focusing you have above H1 and the defocusing  you have above four ninths. But in the focusing, you have nothing and we also wanted to close that  gap. And both of the results follow from a method that we like to call propagation, probabilistic  propagation of regularity. So, so let me put this in here because this is this what I just, I just  said. This is, so let me just talk about the 2D. Why is this not trivial? The result is not trivial  because if I give you data, which is a smoother than the random data that is sitting in H minus epsilon,  that data is with probability zero, I told you, with probability zero is in the support of a measure.  So I cannot just take a smoother data, a larger decay and use Boolean. It doesn't, you have nothing.  Okay? So, so, so, so you usually, when you prove these results, I mean this is what is it that  you do. Having in dispersive PD, having more regularity never helps you. Usually what you do is  you say, well, you need a concert quantity to prove something global. You prove it at the level in  which you have a concert quantity and then you use propagation of regularity to get the smoother  solutions. And here that doesn't work. Okay? And so what we do instead is we sort of, the key idea  is that what we want to do is we want to decompose the data into, into, into a term that is close to  the support of, and I'm going to be made up precisely, close to the support of the measure  and a rougher, in the rougher topology, in the rougher topology that where we are. So, so for  example, in 2D we want to do it for S bigger than zero. So the rougher topology will be the H minus  epsilon and then a smoother reminder. And the question is how to do this so that things work.  And then we want to do sort of like a, sort of a perturbation argument to conclude. Okay? And,  and the argument that I'm going to present here is, is actually we do it in these two cases.  But it's very general. So every time that you have proved a result, that is almost sure using a  measure and, and there is a gap above you, you can use it. So, so every time that you have a  measure, you can use this every single time. So let me just give you this example. So this is the  statement of a theorem. Sorry for it. I have to state something. So u alpha just means it's just  to remind you that the data that I'm taking has this form. Okay? So this alpha means that this  is the data. So you see when alpha is zero, you are in H minus epsilon. And this is with  probability zero in the support of the measure. So u alpha just means that that's the data and you  solve the cubic equation. And I'm above l two. So I don't, even if you normalize or renormalize  your above l two, so you can always put the, the weak ordering, which is finite now into the linear  term. So, so I'm not going to worry about that. And so the theorem says that if you give me any  large t and any alpha positive and some, some epsilon, then there exists a set of, so this is  the Gaussian measure. So there's no Gibbs measure, but you always have a Gaussian measure associated  to the one plus alpha regularity, which is that means that you're, you're forming this with the  derivative to the, to the alpha square instead of, instead of the gradient, you put the gradient one  plus alpha square and the support is here. And so there exists a set with, with measure one such  that you can continue this to, to a global solution and it has this form. Okay. So that's, that's the,  the global result. So how do we prove this? You take the data and it's two steps. So in the first step,  all we want is to, to, to be able to show that this data gives rise to, to a global solution in  some rougher topology. Okay. And then in the step two, we will, we will optimize the regularity.  So the idea is that you take this data and you, you do, you decompose it into, you take some L  that is going to be determined later and you break it into N less than L. And then you take a second  part where you put is like, if you wish, you can think of this AN as a, as a high, as a high pass  filter. Okay. So you let, you let just this, this high frequencies pass. And that's what I call AN.  And you write this as HL plus psi one omega. That's this term. And so what you know is, you know,  this is just the low frequencies then in the, in the H epsilon one, which is the regular, then  this is uniformly bounded for all. Okay. That's no, no, no problem. And, and this AN else are,  are this, as I said, this sort of like high pass filter and in size, they're always less or equal  than one over L. And so now what you do is you use Bourguin's result in H minus epsilon to claim  that there exists a set with probability one such that if you look at now the YL, this YL,  leave this alone, take this YL. Okay. And you add to that this term. Okay. So this is a smoother.  So the probability, the Gibbs measure sees this with probability zero. And this is in the support  of the measure beta L is just one over L. So this is in the support of Bourguin's measure. And you  can add anything smooth to that because this is not seen by the measure. So this is still in the  support of the measure. And so this is one of the elements for which Bourguin can, can, can continue  globally. Okay. So write that as your YL plus an outside two. And so now you know that this evolves  by Bourguin's result. This evolves globally to some solution UL that has this form. And you have  this bounce because Bourguin proved that this is always a smoother than you start. You start here,  you end up here. So furthermore, actually in part of the argument, another one to get in the details,  what you get there is this, in this iteration argument, this approximation and this iteration  argument that I mentioned before, the, the, you get as a sort of byproduct of that you get,  so these are just the XSB norms with epsilon and B equals to a half. So you also get this uniform  bounce on, on these terms in these XSB norms on each iterative time step. Okay. Uniform in L.  So that's part of this approximation and iteration argument of Bourguin that I showed you before.  So now what you do is you use this, that this is global, the UL is global, to prove the existence  and uniqueness of the UAlpha that you want. Okay. And, and the key, the reason that's the,  that's the perturbation argument. And the reason why the perturbation argument works  is because the, these norms are less than one over L. They are small. So these terms are small.  They can be made small. Okay. And so what you do is instead of solving for UAlpha, you look again  at a, at a different, as a suitable difference equation. So you call zeta prime L to be the  linear evolution of what you want minus the solution. This is the, this is what you want to  solve. So you want to solve for that. But instead of looking at that, you look at the difference  between the Bourguin one and that one. That's the perturbation. That looks like this. This,  this solves this problem. And this is kind of like the cubic term. And this is the cubic term.  And the fact that these two things are small is what allows you to, to, to finish with a  contraction and iteration. I mean, that these two things are small plus this uniform bound  allows you to prove that indeed this exists. And then the solution, the z alpha that you're  after is simply this z prime. And so you found, you found the solution as the linear evolution of,  of that data plus the z alpha. And once, yes, yes, once, no, no, no, he's cheating.  Yeah, yeah, he's cheating. No, no, no, no, no. Come on. All right. So, and once you have that,  this step two is the, is the, is the, is the recovery of regularity. Is the recovery of  regularity, which actually I'm going to skip since, since now I'm being pressed. So the second part  is, is what I want to talk is, so all of this was about, so I don't have this console. So the  first part is about not the cosmic censorship, whatever, is about the global repositions which  you don't like. So, but that gives measures which kind of is kind of like the, the, the, the  equilibrium ones. And so what I want to talk about quickly now is about this other result,  which is actually the transfer of energy. This is related to the talk of Patrick, maybe the first  day for the halfway, but also to some lectures that's a her, I mean, it's kind of the other side  of what's a her was describing last week. And so this is kind of like the, the, the typical question  of the transfer of energy of the out of equilibrium dynamics for NLS. And so everybody knows that if  you are on RDU and if you have scattering, there is no sort of like a skates and that on compact  domains you lose that. Okay. So, so the question was how to, how to, how to capture that. And this  is what also goes by the name of turbulence. And Burgen's approach was that what you want to do is  you want to study the growth of the, of the sovereign norms to see how this moves, how this  is transferred. Okay. So he's, he's, he's actually his conjecture in 2000 was whether he asked whether  there exists global solutions to the cubic equations whose Hs norms growing definitely. Okay. This  is what is called the sort of infinite cascade conjecture. And so Burgen himself did some progress  on that. He constructed some special one D non-linear wave equations, some modified equations  in which he, he exhibited that there's some work of Kuskin. And then the fundamental progress came  in this paper of Koleander, Tils, Tafilani, Takaoka Tao in which they actually constructed large  but finite growth on the sovereign norms. And if you haven't seen that result, I put it here.  There's also work of Honey, Gerard, and Prelier, Guarde Kaloshin, Prostitius. And, and more recently,  actually in a really nice paper of Honey Posaders at Govind, Vigilia, sort of they prove that if  instead of the torus you take the product domain, these cylinders, then actually they, they, they,  they solve the conjecture. They said they prove yes and they also gave some, some rate. And on Monday,  the first talk of the, of this conference, we saw Patrick talking about similar things for the,  for the halfway on R. And so now to understand Burgen's conjecture for the cubic analysis,  actually very, very hard, very hard, of course, very hard problem, very ambitious. And so,  but something that which is kind of like in between Burgen and what I described in the first part of  the talk, which is this, this, this invariant Gibbs measures is really the, the, the study of  the existence and uniqueness of non-equilibrium invariant measures. Okay. So this is kind of  like the, the, in the stack mech, is kind of like this is very hard and very poorly understood the,  the, the existence of stationary non-equilibrium states. And, and for us this, this is trying to  maybe give some sort of justification, some sort of like rigorous sort of framework for,  for the statistical description of the sort of auto-equilibrium dynamics. Now this problem is  very hard and it's very hard even for stochastic-califorced systems in which is actually, is actually  very hard, but there has been progress in this sort of like chains of oscillators of Eichmann,  Piller, Reivelle, Reivelle, Thomas and more recently of Hermatingly in which they look at  sort of a collection of unharmonic oscillators with, with nearest, near-neighborhood coupling. So here  you have the Hamiltonians they consider and this is in the first two papers, this is the relationship  between the pinning and the coupling and this is the much harder case in which the relationship is  like this and they're already, they cannot solve this problem unless they only have three oscillators.  So here there is no, no, no sort of problem with the end but here they cannot do more than three.  And the idea is that what you do is you take this and you put them into contact with two heat baths  at different temperatures. So you have your system and what you do is you want to inject energy  into the first mode and dissipate it into the last and you want to try to see whether the system  will relax into, into a stationary non-equilibrium state. And so for us, so, so for us the starting  point is actually this toy model in the work of Colliander-Klisthafilani Tagocca Tao whose Hamiltonian  has this form and, and it is known that proved there. So here this is much harder than the other  one because the interactions don't depend just on the relative distances but also on the momenta  of the particle and the neighbors. And there's known that there is a finite set for which the  resonant NLS is close and collapses to, to this toy model, the one that has this Hamiltonian.  And so we do the same thing, we attach two heat baths at the low mode C1 and Cn and we just sort  of a standard way of, of adding and dissipating energy. And what we expect is this system to,  to converge to a non-equilibrium invariant measure which has some sort of flux from low  frequencies to hard and we also want some sort of like rate of convergence. And so that's,  that's what we want to do. And so the stochastic of the model looks like this for that H. So this  is, you see the heat baths here and we also add these terms, that's the Hamiltonian. So we leave  the middle terms, the middle modes alone and we just do this. And so you look at the, so just,  I wrote it just for convenience, this is the, the sort of the, the generator of the, of the  transition semi-group, the, the Fokker Planck operator is here. And so what you do with the  first thing that you can prove is that if you are at equal temperatures of T1 is Tn, then this  system indeed relaxes to an invariant Gibbs measure. Okay. And all, all of that follows from the fact  that if you look at the adjoint of the, of the operator, Fokker Planck on this, this is zero.  And so what is interesting is in the non-equilibrium case, that's a sort of equilibrium,  is what happened at different temperatures. Okay. And you want to prove whether there exists a  unique smoother-guiding non-equilibrium invariant measures that has this transfer of energy.  And the answer is that again, that if you have three modes, so in other words, we only have  one, two, three, so J is two. Then actually we can prove that and maybe to finish because I  don't want to push your patience. What does it have? Thirty seconds? No, no, it still has three  minutes. Oh, three minutes, perfect. Wonderful. So is that, what I want to explain is that the  proof of this, what is hard is, is the only thing that is really hard is the existence. Okay.  Everything else kind of is a standard. Okay. But the existence of the measure is very, very hard.  So to understand this, let me show you two transparencies. First, first I'm going to maybe  show you what we can prove. So we can also get some rates. And the problem is that we're still  finishing. There is one region which is a little delicate, so we don't know if we are going to get  exponential or polynomial yet. So one useful thing is to change coordinates. Then you have  to change coordinates more than once, but you can change coordinates. So this is the ij, so this is  m. It's the sum of these squares. And this is the angles. And so the first problem we have,  unlike the previous works, is that we lose the hyperliphticity in all of phase space.  So we are not hyperliptic everywhere, but we have to remove this. Okay. We have to  remove, we have a degenerate line that we have to remove. The good thing is that if you start  with c2 equals to zero, then you will remain at c2 equals to zero. But in the hyperliphticity,  which is what gives you that if something exists is going to be smooth, then we have now a boundary.  We have this boundary. And so the existence of the measure follows from what is it that you  want to do. What you want to do is you want to construct the elliponopfunction that somehow  penalizes this region where this is a small and sort of high frequencies. And somehow what you  want to do is once you construct it, then that construction will give you an upper bound on the  heating time of the good region, which should be a compact set. So let me show you quickly a picture,  because maybe then it will be easy. This is actually what is how about the problem. We have to chop  phase space, which they didn't have to do before. And so now it's like this is the good region where  you want all your dynamics to happen. And so what you want to do is you want to prove, you want to  construct the elliponopfunction here that tells you that as soon as you enter any of these regions,  you are kicked back into here. Okay. But usually, you know, usually, when these things are very  hard to do, and usually people what they do is they say, well, you know, your guess, your best  guess is that the elliponopfunction should be e to the m. But that doesn't work everywhere here.  And you have to actually understand very well what are the dynamics of the deterministic system to  understand what works where. And in the regions in which you have more than one boundary, these  boundaries are determined by the temperature and the gamma and so on. But in the regions in which  you have these boundaries, they have contact, they talk to each other, you can guess one and then  the way to solve it, the way to solve it. So you can guess one of them. And then what you have to  do is you have to solve sort of a Poisson equation with the right boundary values and some convexity  conditions and solve for all the other ones, flow it all the other ones to get them. And in some of  the regions to get these, the problem is that you have to really understand to get this to really  have to understand how the phases, you need to understand the phase diagram and know that the  phases will look at some point. And so it's actually quite delicate. And so this is where the  heart of the matter is. And as I said, once you have that, then the rest, the uniqueness and the  ergo-dicity follows from sort of a control argument, a controllability dilemma that tells you  that the deterministic system sort of can access any region in phase space. So if you give me two  points, I can find an epsilon, a neighborhood of one. So that I start here, I end up in the other  one, anywhere in phase space. And then there is a standard theorem of Strucambara that tells you  that once you have that for the deterministic system, the stochastic one will behave the same way.  And thank you very much. I'll leave you the picture. I spent a lot of time on that.  Thank you. Questions? Yes. So, did I understand correctly that the difficulty is to understand  the dynamics of this ODE in big dimensions in a very precise way? Yes, and we cannot do, you see,  the problem is look at this. We have to break all of this only with three modes. If you have four,  for example, what is the problem? The problem is, okay, higher energies, you can do, but the problem  is at low energies, like when you're zero. So if you are stuck in the middle, your energy doesn't  move anywhere. I mean, if you have four modes, maybe you are stuck there. And so, you know,  we are happy first if we can do three. I mean, the paper of Jonathan with Martin, they're also on  three. They cannot do four either. So it's not clear. It will be much more complicated. The way in  which you will have to chop and understand is the construction of this Lyapunov function is hard.  But does it help if you have more conserved quantities?  I don't know.  You see, the Hamiltonian is here. I don't know. Good question.
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All right, so let me first thank very much Frank and Jeremy and Tomás for the invitation to speak at this conference, but also to the other organizers for organizing such a great program and it's great to be here. So what I'm going to talk about is kind of a little bit the plan is and I apologize to some of you who have heard me at least once and I can see Alex there. So like, Michael said, feel free to prove a lemma while I speak on your own work. So anyway, so what I would like to do is because of the mixed audiences to first maybe review a little bit some of the work of Burgen, which is in the background of all of this, and they sort of talk about some recent work with Gileola Stapilani for the cubic equation in 2D and the quintic focusing in 1D and then sort of like talk about some more recent work, which we're still writing with Sacherhane and Jonathan Matty and me look really at Gileola and maybe perhaps between 1 and 2 and 3, I want you to think about like, it's kind of like talking about sort of in the first two is like the notion of Gibbs measures, which is this sort of like equilibrium states come in and while the second part is about sort of like the non-equilibrium states. So also, okay, so let me quickly so I want to talk about the Schrodinger equation and just for the purpose of this talk, I'm going to talk most of the time about the periodic case and I'm going to talk about the square torus, not the rational one. And sort of for the purpose of this talk, every time that I talk about critical or supercritical or subcritical always mean relative to scaling. And so as everybody knows here, because we have people here who have done that work, there has been a lot of progress in studying these equations and most of it has been sort of in deterministic aspects of the phenomena. But there's still some questions that remain. There's essentially like, you know, supercritical case, which even in the defocusing case people don't know if you have blow up. But also there are gaps in the in the well-posed theory, as we will see. So here is just quickly some reviews like of some of the work that you all know on RD. And so this is for subcritical regimes here, the work of Bergen, the work of Colliano-Cristafalian and Taco-Catal. And then we know sort of for critical equations, I mean, and I'm not putting all names, but there has been a lot of like fantastic work and also some conditional results that sort of intercritical. And so and supercritical, as I said, there is not much, there's not nothing known really in the defocusing case. Okay, so on the periodic case, though, the situation is much more limited and the results are fewer. And there's even a problem at the local well-posed theory, right? So Bergen proved, like in 93, that actually dispersion in the periodic is indeed weaker. And so for example in 1D, the L6, so the L4 in 2D actually do have epsilon derivatives. This is actually, he constructed the examples. And so what that means is that in the local well-posed theory, you cannot, although this equation, for example, the cubic in 2D or the quintic in 1D are L2 critical, you cannot close them. There is no local theory at the level of L2. That's open. Okay? S is bigger than zero. There are no results in L2. Local. Okay? And so in the same paper, the subcritical regime, so Bergen sort of proved, as I said, the local well-posed. And these are just some examples. And of course, you have global at the energy level. Okay? So how about global? Well, global, you have the work for, so there were work for small data, for Tess and Cofftetaru. And then the sort of the breakthrough work, the first sort of like global, large data global of Jonas Komposadere, who actually did the large data for the quintic defocusing in 3D. A student of mine did, after the breakthrough, actually, of Bergen, the matter of the strikers, he did, that's part of his thesis, he did the cubic defocusing in a lesson 4D. And as I said, there was a small data result before. And maybe I should mention here that there's work of Kilipandisan, which they have also a different proof of the small data and new proofs for the irrational case. But as I said, I'm going to focus on the rational case. And the reason is because the non-deterministic setting, you still rely heavily on analytic number theory. And so we don't know still how to treat many of the things in the rational case. So the point is that, as I said in the L2 level, this is not even known. So the point of view that you want to take is that then maybe what you should do is you should study these equations from a non-deterministic point of view. And this brings us to Bergen, which is the starting point in 1994-96, where he studied long-time dynamics for periodic equations in the almost sure sense, and showed that you have actually global well-posedness on a set of data, a full Gibbs measure. So the Gibbs measures for NLS have been constructed, for example, for the cubic NLS in 1D, 2D, and 3D defocusing, have been constructed by Glimm-Jaffin, the context of quantum field theory, like the P4 model, or the Stochastic Allen-Kahn. And then in the context of Hamiltonian systems by Leuvelius-Rosenspier, who also treated the focusing case in one dimension. Okay, here you have, if you don't know, that just in variance means that you have this measure that if you take a set of initial data, as you flow it, it preserves the measure. And so why is it that it works? I mean, because usually what happens is, when you are sitting on the, and I'll show you in a minute, when you are sitting in the statistical ensemble, or in the support of the measure, you are at a regularity which is very rough, and there are no concert quantities at that level. So why is it that you can get global? Okay, so the point is that, assuming that you have a local well-posedness, and it doesn't have to be deterministic, it could be non-deterministic, then the invariance of the measure where it acts as a concert quantity that allows you to continue the results from local to global. At a level of regularity where there are no concert quantities, so that's the key. Okay? And the virtue of doing this is that somehow it captures generic behavior, and I'll say something more in a minute about that. And so now this, yes, I will explain it a little. So an idea of what do I mean? And so, I mean, generic in the sense of probability, that's what I mean. Okay? But I'll, you know, infinite dimensions is a big place. But, you know, all right. So actually there are some, of course, limitations and challenges. One is actually for this perceived PDE, the actual constructions of these measures. They are not that easy. They're easy to construct if you are on finite measure, on the domains, if you're in infinite measure or infinite volume, there are some problems. If the PDE is not Hamiltonian, it's not always clear how to do it. And as I said in 2D and 3D, in the focusing, you actually, you cannot construct it directly, it was proved that actually what you have to do is you have to renormalize the nonlinearity and remove some sort of like divergences of some infinities in order for the measure to make sense. That was known. In the focusing case for 2D, Bridges and Slate said that there was no Gibbs measure. The measure is not available in higher dimensions like 4. And even in 3D, if instead of cubic, you are quintic, then it's also known that if you try to do the same renormalization procedure, you either go to something trivial or to the Gaussian measure, so there are no Gibbs measures either. And for example, if you take something like the defocusing quintic in 2D or the cubic in 3D for which there is a measure, the problem, and I'm going to show you why in a minute, is that in order to prove the invariance, you of course have to have a global flow and you can have actually a weak solution. But if you want to show the global well postness, then you are going to see it as you go high in dimensions, you're going to see it in rougher and rougher spaces, and it's actually not clear how to do this. This is challenging. The defocusing quintic in 2D might be within reach thanks to the new techniques from some sort of like the baradiff... baracontrol distributions for guvinelli or regularity structures, but this is a little too soon to talk about that. So quickly, how is this measure defined? Because I'm going to be needing this, so of course you think of your equation as an infinite dimensional Hamiltonian system on the Fourier coefficients. And let's say one dimension, let's take just one dimension, p less than equal to n5, we'll also have Spear said, well, you can construct this measure that formally looks like this. So this is like Lebesgue measure, this is Hamiltonian, and this is the measure. Now if you look at this, this is total nonsense, because there are three terms here, the Lebesgue measure and the kinetic and the potential energy, and every single factor is infinity. So this makes no sense, but it's a justive of what is it that you want to do, so you cannot define even the Lebesgue measure. But what you do is you construct it using Gaussian measure as a reference measure, and then you construct the Gibbs measure as a weighted measure in two steps. So the Gaussian measure on some sort of like Heeler space, whatever you wish, you construct it as the weak limit of finite dimensional measure. So here you see this is the kinetic part of your Hamiltonian, and this is the Lebesgue measure, which if you're Hamiltonian, the Lebesgue theorem tells you there is volume preserving. And then this raw ending is sort of like you can view it as an induced probability measure under this map, okay? So where these are just from now on, these are always IID complex Gaussian random variables taken with mean zero. And so the point now, the most important point now is that you want to understand the Gaussian measure raw as a weak limit of these finite dimensional approximations, and it's easy to see that they are additive, but they are not countably additive. And so the point is that if you want them to be countably additive, and there is a theorem, a trace theorem that tells you when they are countably additive, then the point is that on HS, they are countably additive only if S is less than a half, okay? But not in Sb or equal to a half. They are in 2D, they are countably additive if you are below L2, with probability zero more. In 3, if you are below minus a half probability zero, if you are above, okay? So what that means is that if I want to, this is exactly what guides you, what is the support of the measure, and where is the data going to leave? That's the regularity of the data you're going to take. So in 1D, if you're looking at the Gaussian measure, you have to be sort of in H a half minus. If you are in 2D, you have to be just below L2 and so on. So you see as you go high in dimensions, you get very, very rough. And so once you do this, then you know that this Gaussian measure is just the law for this random variable. And so these are the typical elements in the support of the measure. This is how they look, okay? And so this is a function almost surely in 1D and in distribution, in higher D, and you have these tail estimates. So as I said before, once you have constructed the Gaussian measure, then the Gibbs measure is just a weighted measure where sort of this is the nonlinear part of the Hamiltonian, this is a renormalization. And what you want is this to be sort of the Radon-Nicholim derivative. You want this to be in L1 relative to rho. And so if you are in the focusing case, you have nothing for free because you still in order to be, to do the renormalization for this to be probability measure, you have to put the L2 cutoff, okay? So the same cutoff that you see in the deterministic case comes here. And actually it's a nice exercise to see which constant is larger. And actually this one is smaller. So you don't win anything. But anyway, so if the power is less than 5, you just need some cutoff. And if the power is 5, which is the critical, mass critical, then it has to be sufficiently small. All right, so what is that Morgan D? It's like Morgan proving 1D. This is his result. So I just, I'm just stating it because I want you to get used to how he reads, is that you take the equation with that data, which is the data in the support of the measure. And then what does it mean to prove all more sure? It means that you can find a set in here. If you are focusing, you have to condition everything to have a small L2 of measure 1 of the, of the gives measure 1, such that for any such data, the initial value problem can be continued globally, okay? And then you also prove that this measure is also invariant, okay? So why is this important? Well, let's recall for a second, deterministically, you have local, if this is strictly bigger than zero, and globally, if you are in H1. And, and that's it. And so what the invariance measure gives you, as I said before, is a global result in H1-2, where there is nothing, there's no conserved quantity. So how are you going to get it? But that's, that's what the, the measure does for you. And so how does he prove this? Let me just quickly go through this, because we are going to need it. It's like, what, what you have to do is you have to approximate the equation by a finite dimensional approximation. And, and, and actually the crucial fact, because what I want to bring home here is, is how do you use the invariance of the measure? And what is important, which is a little bit of a, of a, of a confusion is that what you, what you really use is the invariance of the finite dimensional measure, not the, not the invariance of the infinite. Variance of the infinite sometimes comes in parallel to the, to the global well-posedness at the end. So, so why is it that the finite dimensional measure is, is, is invariant? Well, because for this equation, when you take these things, you stay Hamiltonian. So this stays Hamiltonian, which is not always true when you take an equation and in order to prove a local result, you have to gauge. I mean, what happens is that you might stay Hamiltonian, but the problem is that you don't know how it looks. And this is actually a very delicate point. What happens with Gaussian measures or measures under sort of these non-canonical transformations. And then as I said before, you will see it and then tells you that because you stay Hamiltonian, this is volume preserving, this is conserved. So this is a conserved, this is, this measure is, is invariant. And now what you do is you use the invariance of the finite dimensional measure to continue. So you know, by the, by the local theory, you have a deterministic local theory for this local. And so now you use the invariance of this to continue the solution to the finite dimensional approximation to a global one in time, provided that you have, you are on a set of good data. So what that means is that in this process, you have to select a set of, of omegas with probability one that tell you that you're still, that you can continue, that you can do this process. So once you have this, then how do you continue? You, there is this approximation lemma that tells you that once you have the finite dimensional reach time t, you take the u that you know that is local and you walk the u in an iterative state, state, a method side by side with the u n. And this is not a trivial problem because what happens is that you see after the initial the local time of existence, you somehow this two can become really far apart. Okay. And you have no a priori bound on u. So this is not trivial lemma. So what you have to do is you have to take as a stepping stone, an intermediate system, which is the infinite dimensional equation with projected data. And then what you do is you, you, you use that as a stepping stone and you compare u to u prime and then you pride to u n step by step. And then that's, that's how you do it. And so once you have that, now you have now the, the sort of the global you and sort of the icing in the cake is that you get the, you prove that you have these converges to, to, to mu and you prove you get the invariance of mu. But it's really the invariance of, of, of mu n. Okay. So how about 2d, what did Morgan did in 2d and here is where it becomes interesting. Maybe what I want to say is that so in 2d he considered now the cubic, which is still a little too critical. But now, as I said before, the Gibbs measure, which was known to exist, is, needs some sort of like renormalization in order to exist. Okay. And you can actually see this renormalization also in trying to prove this theorem because it's actually the same thing you need to remove to deal with resonances. And so what you do is you have to treat what is called the weak order nonlinearity. So you have to subtract this term from the cubic. And then he proved exactly the same theorem that then your globally will pose below L2. And the associated measure is invariant. So here, now this is the point that in the, in the kinetic, in the potential energy, this is unbounded almost surely. So you have to sort of weak order and so you renormalize the finite dimensional approximation and, and that removes this term. And what I would like to say is that actually this is the first result ever that is global and supercritical. Okay. So in the, in the dispersive community, this is the first result that is supercritical and global. So is it critical in the sense of L2 even? Yeah, scaling. Every time, as I said, every time I talk about critical, supercritical is relative to scaling. Okay. And so, and so something nice that is actually, just an observation is that you see that the measure sort of always charges open sets with positive measure. So if you prove using this argument, if you can prove that such a result with, with probability, with measure one, then what that tells you is that you cannot have blow up, blow up in that topology that is, you can have blow up, but you cannot be stable in that topology. Okay. So that's actually something nice. Okay. Because, you know, okay, you can still have this is the focus and you don't know if you're going to blow up, but it cannot be stable. Okay. So that tells you that that's kind of like this. Okay. So now the additional difficulty here is that Burgen did not have a local well post in place as he did in 1D. Okay. So he cannot even start the argument that I just described a minute ago because there's no local, there is no deterministic local well post in this before below L2 is supercritical. So what he says is, he says, well, look, I mean, I just want to prove a normal sure global result. So why do I need a deterministic local? It's enough to prove that I have a local well postness for data that is in the support of the measure. In other words, all I need is a probabilistic local well postness result. And so if you look at this paper from 96, which is a very long paper, it's like 56 pages, like, you know, 50 of those are just to prove the probabilistic local well postness. Okay. So that's, that's, that's the, that's the heart of matter in that paper. And the rest just follows the same as before. So that's what I said there. Okay. And this is what I said. I mean, so this brings us to this notion of saying, okay, so now forget about for a moment about the use measure and forget a moment about global. Let's think about just the local well postness. And so what, what Burgen is saying essentially is that if you, if you, if you, if you take data and you randomize the data, then you're always going to be able to improve your local theory. Okay. So that's the takeaway. Okay. And this is actually not a foreign idea for, for, for us and for, for anybody, but, but especially for harmonic analysts, because this is exactly what, what, what, what, what we know and we use all the time. When you prove little pale inequalities, for example, that sort of random series enjoy better LP, LP estimates. And so this randomization doesn't improve the regularity. What it does is improves the LP, the, the integrability. But once you have that, then you use that in turn to get better estimates that you would otherwise deterministically. So what is his strategy to prove this? And this is actually the same strategy that I don't know if at the same time, but, but in the time, this is the same strategy but what is used, used by the Pratt and the Bush for the, for the stochastic Alencona for the FIFOR model. And they were using exactly the same idea. What's the idea? The idea is that now you look solutions of a particular form. So what you'd say is you say, well, you randomize the data now what I'm going to do is I'm going to look for solutions that look like the linear evolution of the random data plus some other W. And instead of solving for you, you solve for the difference equation. Okay. And so as a consequence of this, and so what he can prove is that if you solve for this difference equation, now you solve here in a smoother space that you start. Okay. So you solve for a fixed, you know, whatever you want to do, but you're going to find the W in a smoother regularity than the initial data. And so as a consequence, he proves almost surely in omega that the nonlinear part always is a smoother than the linear part. Okay. This is in this 96 and 94 or 96 paper. Now an important remark here is that by doing this, it's not that you're just saying that if you randomize, you bring the problem from being like super critical to being subcritical. That's not true. I mean, what it is is that you have a problem where the nonlinearity is sort of hybrid, right? I mean, you have supercritical terms which are random, and you have sort of like W's which you treat deterministically, which are really maybe smoother or subcritical. But you have to treat all interactions, random, random, deterministic, deterministic, random, deterministic. And so one of the things that you need in place before you even start this thing is because you have to deal with the deterministic, deterministic at a smoother space, you actually have to have in place a deterministic local well postness in a smoother space to start, at least in this approach. All right. So let me just quickly tell you why it works, so how it works. So everybody, I'm sure here just stated by everybody knows sort of the typical large deviation estimates. So think of k here as being how many random terms you are dealing with in your nonlinearity, whatever that is. And so here you have just a sort of a linear combination of the random variables. So a large deviation tells you that in omega, sort of the LP norm is controlled by the L2 norm. And so what's the point of this? The point of this is that then by Chevich's, the set where the absolute value of this function is bigger than lambda is the case exponentially. So now take any delta and take lambda here to be delta to the minus k over 2 times the L2 norm and put it here. And so what that tells you is that there exists a set, okay, which is the complement of this. So this omega is the complement of this. So the probability of the complement of omega is the probability of this, which if you put this number here gives you that. That tells you that if you're outside that set, right, so if you're outside this set, so that means if you're here, then you can replace the absolute value of this by the L2 norm, square. And that's fundamental. So let me show you why. Okay, that's the key. So what's why? So let's do one example. Suppose that you have data that looks like this, okay. This is what we're going to have if you are in two dimensions just below L2, it will look like this, okay. And in doing your random-random interactions, what you have to do is you have to estimate things of this form, okay. Where this is just the set of n1 and 2 and 3, each of them in Z2, such that they behave like this. Now what the wake ordering does for you, which is actually very interesting because it's a different way of understanding resonance, is that it tells you that n1 and n3, these two are never n2, which means that you never lose the independence of the random variables. So every time that n1 is n2, this becomes an absolute value square, you lose the independence. So the wake ordering, what it does, which is the same as the resonance, it tells you that this stays, you always have independence, okay. And so now if you just were to do Cauchy-Schwarz here, if you want to estimate the absolute value of this, sorry, the L2 norm of this, and you look at this, and you simply were to, so you sum in n and in n because I'm doing L2, and you simply do Cauchy-Schwarz, then you pick up, you lose because you pick up the cardinality of that set. You pick up the cardinality of this set, which is big, okay. It's big. These are integers. You're counting lattice points on spheres or intersections of spheres or whatever. And then you lose, okay. And what that loss means that you cannot lose derivatives, you cannot close. And so what the large deviation estimate tells you is that you are replacing, as I said a minute ago, modulo, a set of omega, you can replace the absolute value with the L2. And so what you are looking at is at this, without that. And that's enough to close, okay. So that's how it works. I mean, that's how it works in the old run, okay. Now, I want to say that something just because Sergio's here, he was just saying quickly, that actually, well, maybe I'll say it in a minute, okay. Let me actually move a couple of transpositions. So the takeaway from here is that the randomization, what it does, is actually allows you to improve the local well-pulsed theory. And you should view the randomization as a separate issue from the invariance of the measure. Once you have improved your local well-pulsedness, the question is how do you go to global? And then it depends which equation you have, which regime you are. If you have a measure, you don't have a measure, you can actually do a variety of things, or you may not be able to do anything, at least with current technologies. At least nothing that is conditional. I mean, one thing that I should say here, maybe going back, is that one problem that you have is that you see, even if you have conserved quantities, or even if you have conserved quantities for you, in the way that you study this problems, when you study the difference equations, you have no conserved quantities. I mean, you lose, the W doesn't satisfy anything. So it's a bit problematic. Okay, so as I said, then how do you pass, so separate the two issues. Randomization improves local, and then if you have a measure, then you can go from local to global. If you don't have a measure, maybe you can do energy estimates, or you can do something else, depend on the regime, or you are still, you may be still working on it, because you have nothing. And as I said, there was a, this has been a lot of work after Burgen. So nothing happened between Burgen's work, and sort of around 2007. And there was a lot of activity. I'm not being comprehensive here. There's a lot of people that have worked for a lot of equations, and not just dispersive, also sort of like on fluids equations. And maybe because it's just search juice here, I wasn't going to say anything, because it's kind of like, this is actually part of a long term project. But you know, in this paper that we sort of consider sort of an linear wave equation with the climate market, null forms, the QIJs. And actually, this, we are interested in this question for, in this equation for a geometric purpose, and for trying to understand these methods beyond the, this regime in which what you do is you look around the linear evolution. We want to look around special solutions, and this, and these geometric equations have like a sort of ground state. So we are interested in trying to understand how to sort of understand, how to propagate a little bit better, how to understand the transport of the randomization. And, but what I want to say is that, you know, it's actually, it's a beautiful recast of these null forms that actually you can, they're exactly, I mean, they're like a gift, like keep on giving, because they are exactly what you need when you have quadratic derivatives to preserve the independence that I showed you before. So I told you that if you have random, random interactions, you have to do something like the weak ordering to make sure that you, you, you keep independent the random variables. And then, and then, and what the null form does is exactly that renormalization in the probabilistic context. And so, so it's actually quite interesting. And so let me quickly talk about this, this work. So this is actually two things in which, two results which are still at the subcritical regime. But the reason why we are interested in this is because they are actually closing an important gap in the subcritical theory between the deterministic theory and the, and the almost sure global well-posed. And so one is in 2D, you take the cubic defocusing equation. And as I said a minute ago, so what we're going to prove was that if you are below L2, you have almost sure global. And for the look, for the global deterministic using the i method, the best result known, which I have a picture here, is above two thirds. Okay. And so we're going to prove it here. And if you are here, you have global. But as I will explain in a minute, if you have a global well-posed that is a probabilistic, you cannot use just the deterministic propagation of regularity methods to, to, to fill in this gap. And so we were interested in understanding how to get global in this range. And for the 1D, which is focusing, you have the same problem, which there is a still a gap between, so that the almost sure global is just below half. And then of course you have in the focusing you have above H1 and the defocusing you have above four ninths. But in the focusing, you have nothing and we also wanted to close that gap. And both of the results follow from a method that we like to call propagation, probabilistic propagation of regularity. So, so let me put this in here because this is this what I just, I just said. This is, so let me just talk about the 2D. Why is this not trivial? The result is not trivial because if I give you data, which is a smoother than the random data that is sitting in H minus epsilon, that data is with probability zero, I told you, with probability zero is in the support of a measure. So I cannot just take a smoother data, a larger decay and use Boolean. It doesn't, you have nothing. Okay? So, so, so, so you usually, when you prove these results, I mean this is what is it that you do. Having in dispersive PD, having more regularity never helps you. Usually what you do is you say, well, you need a concert quantity to prove something global. You prove it at the level in which you have a concert quantity and then you use propagation of regularity to get the smoother solutions. And here that doesn't work. Okay? And so what we do instead is we sort of, the key idea is that what we want to do is we want to decompose the data into, into, into a term that is close to the support of, and I'm going to be made up precisely, close to the support of the measure and a rougher, in the rougher topology, in the rougher topology that where we are. So, so for example, in 2D we want to do it for S bigger than zero. So the rougher topology will be the H minus epsilon and then a smoother reminder. And the question is how to do this so that things work. And then we want to do sort of like a, sort of a perturbation argument to conclude. Okay? And, and the argument that I'm going to present here is, is actually we do it in these two cases. But it's very general. So every time that you have proved a result, that is almost sure using a measure and, and there is a gap above you, you can use it. So, so every time that you have a measure, you can use this every single time. So let me just give you this example. So this is the statement of a theorem. Sorry for it. I have to state something. So u alpha just means it's just to remind you that the data that I'm taking has this form. Okay? So this alpha means that this is the data. So you see when alpha is zero, you are in H minus epsilon. And this is with probability zero in the support of the measure. So u alpha just means that that's the data and you solve the cubic equation. And I'm above l two. So I don't, even if you normalize or renormalize your above l two, so you can always put the, the weak ordering, which is finite now into the linear term. So, so I'm not going to worry about that. And so the theorem says that if you give me any large t and any alpha positive and some, some epsilon, then there exists a set of, so this is the Gaussian measure. So there's no Gibbs measure, but you always have a Gaussian measure associated to the one plus alpha regularity, which is that means that you're, you're forming this with the derivative to the, to the alpha square instead of, instead of the gradient, you put the gradient one plus alpha square and the support is here. And so there exists a set with, with measure one such that you can continue this to, to a global solution and it has this form. Okay. So that's, that's the, the global result. So how do we prove this? You take the data and it's two steps. So in the first step, all we want is to, to, to be able to show that this data gives rise to, to a global solution in some rougher topology. Okay. And then in the step two, we will, we will optimize the regularity. So the idea is that you take this data and you, you do, you decompose it into, you take some L that is going to be determined later and you break it into N less than L. And then you take a second part where you put is like, if you wish, you can think of this AN as a, as a high, as a high pass filter. Okay. So you let, you let just this, this high frequencies pass. And that's what I call AN. And you write this as HL plus psi one omega. That's this term. And so what you know is, you know, this is just the low frequencies then in the, in the H epsilon one, which is the regular, then this is uniformly bounded for all. Okay. That's no, no, no problem. And, and this AN else are, are this, as I said, this sort of like high pass filter and in size, they're always less or equal than one over L. And so now what you do is you use Bourguin's result in H minus epsilon to claim that there exists a set with probability one such that if you look at now the YL, this YL, leave this alone, take this YL. Okay. And you add to that this term. Okay. So this is a smoother. So the probability, the Gibbs measure sees this with probability zero. And this is in the support of the measure beta L is just one over L. So this is in the support of Bourguin's measure. And you can add anything smooth to that because this is not seen by the measure. So this is still in the support of the measure. And so this is one of the elements for which Bourguin can, can, can continue globally. Okay. So write that as your YL plus an outside two. And so now you know that this evolves by Bourguin's result. This evolves globally to some solution UL that has this form. And you have this bounce because Bourguin proved that this is always a smoother than you start. You start here, you end up here. So furthermore, actually in part of the argument, another one to get in the details, what you get there is this, in this iteration argument, this approximation and this iteration argument that I mentioned before, the, the, you get as a sort of byproduct of that you get, so these are just the XSB norms with epsilon and B equals to a half. So you also get this uniform bounce on, on these terms in these XSB norms on each iterative time step. Okay. Uniform in L. So that's part of this approximation and iteration argument of Bourguin that I showed you before. So now what you do is you use this, that this is global, the UL is global, to prove the existence and uniqueness of the UAlpha that you want. Okay. And, and the key, the reason that's the, that's the perturbation argument. And the reason why the perturbation argument works is because the, these norms are less than one over L. They are small. So these terms are small. They can be made small. Okay. And so what you do is instead of solving for UAlpha, you look again at a, at a different, as a suitable difference equation. So you call zeta prime L to be the linear evolution of what you want minus the solution. This is the, this is what you want to solve. So you want to solve for that. But instead of looking at that, you look at the difference between the Bourguin one and that one. That's the perturbation. That looks like this. This, this solves this problem. And this is kind of like the cubic term. And this is the cubic term. And the fact that these two things are small is what allows you to, to, to finish with a contraction and iteration. I mean, that these two things are small plus this uniform bound allows you to prove that indeed this exists. And then the solution, the z alpha that you're after is simply this z prime. And so you found, you found the solution as the linear evolution of, of that data plus the z alpha. And once, yes, yes, once, no, no, no, he's cheating. Yeah, yeah, he's cheating. No, no, no, no, no. Come on. All right. So, and once you have that, this step two is the, is the, is the, is the recovery of regularity. Is the recovery of regularity, which actually I'm going to skip since, since now I'm being pressed. So the second part is, is what I want to talk is, so all of this was about, so I don't have this console. So the first part is about not the cosmic censorship, whatever, is about the global repositions which you don't like. So, but that gives measures which kind of is kind of like the, the, the, the equilibrium ones. And so what I want to talk about quickly now is about this other result, which is actually the transfer of energy. This is related to the talk of Patrick, maybe the first day for the halfway, but also to some lectures that's a her, I mean, it's kind of the other side of what's a her was describing last week. And so this is kind of like the, the, the typical question of the transfer of energy of the out of equilibrium dynamics for NLS. And so everybody knows that if you are on RDU and if you have scattering, there is no sort of like a skates and that on compact domains you lose that. Okay. So, so the question was how to, how to, how to capture that. And this is what also goes by the name of turbulence. And Burgen's approach was that what you want to do is you want to study the growth of the, of the sovereign norms to see how this moves, how this is transferred. Okay. So he's, he's, he's actually his conjecture in 2000 was whether he asked whether there exists global solutions to the cubic equations whose Hs norms growing definitely. Okay. This is what is called the sort of infinite cascade conjecture. And so Burgen himself did some progress on that. He constructed some special one D non-linear wave equations, some modified equations in which he, he exhibited that there's some work of Kuskin. And then the fundamental progress came in this paper of Koleander, Tils, Tafilani, Takaoka Tao in which they actually constructed large but finite growth on the sovereign norms. And if you haven't seen that result, I put it here. There's also work of Honey, Gerard, and Prelier, Guarde Kaloshin, Prostitius. And, and more recently, actually in a really nice paper of Honey Posaders at Govind, Vigilia, sort of they prove that if instead of the torus you take the product domain, these cylinders, then actually they, they, they, they solve the conjecture. They said they prove yes and they also gave some, some rate. And on Monday, the first talk of the, of this conference, we saw Patrick talking about similar things for the, for the halfway on R. And so now to understand Burgen's conjecture for the cubic analysis, actually very, very hard, very hard, of course, very hard problem, very ambitious. And so, but something that which is kind of like in between Burgen and what I described in the first part of the talk, which is this, this, this invariant Gibbs measures is really the, the, the study of the existence and uniqueness of non-equilibrium invariant measures. Okay. So this is kind of like the, the, in the stack mech, is kind of like this is very hard and very poorly understood the, the, the existence of stationary non-equilibrium states. And, and for us this, this is trying to maybe give some sort of justification, some sort of like rigorous sort of framework for, for the statistical description of the sort of auto-equilibrium dynamics. Now this problem is very hard and it's very hard even for stochastic-califorced systems in which is actually, is actually very hard, but there has been progress in this sort of like chains of oscillators of Eichmann, Piller, Reivelle, Reivelle, Thomas and more recently of Hermatingly in which they look at sort of a collection of unharmonic oscillators with, with nearest, near-neighborhood coupling. So here you have the Hamiltonians they consider and this is in the first two papers, this is the relationship between the pinning and the coupling and this is the much harder case in which the relationship is like this and they're already, they cannot solve this problem unless they only have three oscillators. So here there is no, no, no sort of problem with the end but here they cannot do more than three. And the idea is that what you do is you take this and you put them into contact with two heat baths at different temperatures. So you have your system and what you do is you want to inject energy into the first mode and dissipate it into the last and you want to try to see whether the system will relax into, into a stationary non-equilibrium state. And so for us, so, so for us the starting point is actually this toy model in the work of Colliander-Klisthafilani Tagocca Tao whose Hamiltonian has this form and, and it is known that proved there. So here this is much harder than the other one because the interactions don't depend just on the relative distances but also on the momenta of the particle and the neighbors. And there's known that there is a finite set for which the resonant NLS is close and collapses to, to this toy model, the one that has this Hamiltonian. And so we do the same thing, we attach two heat baths at the low mode C1 and Cn and we just sort of a standard way of, of adding and dissipating energy. And what we expect is this system to, to converge to a non-equilibrium invariant measure which has some sort of flux from low frequencies to hard and we also want some sort of like rate of convergence. And so that's, that's what we want to do. And so the stochastic of the model looks like this for that H. So this is, you see the heat baths here and we also add these terms, that's the Hamiltonian. So we leave the middle terms, the middle modes alone and we just do this. And so you look at the, so just, I wrote it just for convenience, this is the, the sort of the, the generator of the, of the transition semi-group, the, the Fokker Planck operator is here. And so what you do with the first thing that you can prove is that if you are at equal temperatures of T1 is Tn, then this system indeed relaxes to an invariant Gibbs measure. Okay. And all, all of that follows from the fact that if you look at the adjoint of the, of the operator, Fokker Planck on this, this is zero. And so what is interesting is in the non-equilibrium case, that's a sort of equilibrium, is what happened at different temperatures. Okay. And you want to prove whether there exists a unique smoother-guiding non-equilibrium invariant measures that has this transfer of energy. And the answer is that again, that if you have three modes, so in other words, we only have one, two, three, so J is two. Then actually we can prove that and maybe to finish because I don't want to push your patience. What does it have? Thirty seconds? No, no, it still has three minutes. Oh, three minutes, perfect. Wonderful. So is that, what I want to explain is that the proof of this, what is hard is, is the only thing that is really hard is the existence. Okay. Everything else kind of is a standard. Okay. But the existence of the measure is very, very hard. So to understand this, let me show you two transparencies. First, first I'm going to maybe show you what we can prove. So we can also get some rates. And the problem is that we're still finishing. There is one region which is a little delicate, so we don't know if we are going to get exponential or polynomial yet. So one useful thing is to change coordinates. Then you have to change coordinates more than once, but you can change coordinates. So this is the ij, so this is m. It's the sum of these squares. And this is the angles. And so the first problem we have, unlike the previous works, is that we lose the hyperliphticity in all of phase space. So we are not hyperliptic everywhere, but we have to remove this. Okay. We have to remove, we have a degenerate line that we have to remove. The good thing is that if you start with c2 equals to zero, then you will remain at c2 equals to zero. But in the hyperliphticity, which is what gives you that if something exists is going to be smooth, then we have now a boundary. We have this boundary. And so the existence of the measure follows from what is it that you want to do. What you want to do is you want to construct the elliponopfunction that somehow penalizes this region where this is a small and sort of high frequencies. And somehow what you want to do is once you construct it, then that construction will give you an upper bound on the heating time of the good region, which should be a compact set. So let me show you quickly a picture, because maybe then it will be easy. This is actually what is how about the problem. We have to chop phase space, which they didn't have to do before. And so now it's like this is the good region where you want all your dynamics to happen. And so what you want to do is you want to prove, you want to construct the elliponopfunction here that tells you that as soon as you enter any of these regions, you are kicked back into here. Okay. But usually, you know, usually, when these things are very hard to do, and usually people what they do is they say, well, you know, your guess, your best guess is that the elliponopfunction should be e to the m. But that doesn't work everywhere here. And you have to actually understand very well what are the dynamics of the deterministic system to understand what works where. And in the regions in which you have more than one boundary, these boundaries are determined by the temperature and the gamma and so on. But in the regions in which you have these boundaries, they have contact, they talk to each other, you can guess one and then the way to solve it, the way to solve it. So you can guess one of them. And then what you have to do is you have to solve sort of a Poisson equation with the right boundary values and some convexity conditions and solve for all the other ones, flow it all the other ones to get them. And in some of the regions to get these, the problem is that you have to really understand to get this to really have to understand how the phases, you need to understand the phase diagram and know that the phases will look at some point. And so it's actually quite delicate. And so this is where the heart of the matter is. And as I said, once you have that, then the rest, the uniqueness and the ergo-dicity follows from sort of a control argument, a controllability dilemma that tells you that the deterministic system sort of can access any region in phase space. So if you give me two points, I can find an epsilon, a neighborhood of one. So that I start here, I end up in the other one, anywhere in phase space. And then there is a standard theorem of Strucambara that tells you that once you have that for the deterministic system, the stochastic one will behave the same way. And thank you very much. I'll leave you the picture. I spent a lot of time on that. Thank you. Questions? Yes. So, did I understand correctly that the difficulty is to understand the dynamics of this ODE in big dimensions in a very precise way? Yes, and we cannot do, you see, the problem is look at this. We have to break all of this only with three modes. If you have four, for example, what is the problem? The problem is, okay, higher energies, you can do, but the problem is at low energies, like when you're zero. So if you are stuck in the middle, your energy doesn't move anywhere. I mean, if you have four modes, maybe you are stuck there. And so, you know, we are happy first if we can do three. I mean, the paper of Jonathan with Martin, they're also on three. They cannot do four either. So it's not clear. It will be much more complicated. The way in which you will have to chop and understand is the construction of this Lyapunov function is hard. But does it help if you have more conserved quantities? I don't know. You see, the Hamiltonian is here. I don't know. Good question.
then sort of like talk about some more recent and also some conditional results that sort of measure. So the Gibbs measures for NLS have been if you take a set of initial data, as you flow show you in a minute, when you are sitting in the non-deterministic, then the invariance of the PDE, the actual constructions of these measures. Gibbs measure. The measure is not available in reach thanks to the new techniques from some sort Hamiltonian system on the Fourier coefficients. that they are additive, but they are not can be continued globally, okay? And then you with, with probability, with measure one, then is in the support of the measure. In other words, and the almost sure global well-posed. And so one So how do we prove this? You take the data and not seen by the measure. So this is still in the global, to prove the existence and uniqueness of