Let


* $U,H$ be separable $\mathbb R$-Hilbert spaces
* $Q\in\mathfrak L(U)$ be nonnegative and self-adjoint with finite trace
* $U\_0:=Q^{1/2}U$
* $(\Omega,\mathcal A,(\mathcal F\_t)\_{t\ge 0},\operatorname P)$ be a filtered probability space
* $(W\_t)\_{t\ge 0}$ be a $Q$-Wiener process on $(\Omega,\mathcal A,(\mathcal F\_t)\_{t\ge 0},\operatorname P)$
* $u\in C^{1,\:2}([0,\infty)\times H,H)$
* $G:[0,\infty)\times H\to\operatorname{HS}(U\_0,H)$ with $G(t,\;\cdot\;)$ being Fréchet differentiable for all $t\ge 0$
* $X\_0$ be a $\mathcal F\_0$-measurable random variable on $(\Omega,\mathcal A,\operatorname P)$ and $$X\_t:=X\_0+\int\_0^tu(s,X\_s)\:{\rm d}s+\int\_0^tG(s,X\_s)\circ{\rm d}W\_s\;\;\;\text{for }t>0\tag 1$$


By definition of $X$, we obtain $${\rm d}u(t,X\_t)=\left[\frac{\partial u}{\partial t}(t,X\_t)+{\rm D}u(t,X\_t)u(t,X\_t)\right]{\rm d}t+\underbrace{{\rm D}u(t,X\_t)G(t,X\_t)}\_{=:\:\Phi(t,X\_t)}\circ{\rm d}W\_t\tag 2$$ for all $t\ge 0$. If we interpret $t\mapsto X\_t$ as being the trajectory of a fluid particle perturbed in the sense of $(1)$ and $u(t,X\_t)$ as being the velocity of this particle at time $t$, then it's not far from being reasonable to assume $${\rm d}u(t,X\_t)=\left[-\frac1\rho\nabla p(t,X\_t)+\nu\Delta u(t,X\_t)\right]{\rm d}t\tag 3$$ for all $t\ge 0$ by *conservation of momentum* for some $\rho,\mu>0$ and $p:[0,\infty)\times H\to\mathbb R$ with $p(t,\;\cdot\;)\in C^1(\Lambda)$ for all $t\ge 0$ for some bounded and open $\Lambda\subseteq\mathbb R^d$ for all $t\ge 0$ in the case $H=\mathbb R^d$ for some $d\in\mathbb N$. By equating $(2)$ and $(3)$, we obtain a SDE for $u$. Actually, I want to solve that SDE numerically. However, I want to ensure that a solution (in some suitable sense) is guaranteed by well-known theory (e.g. the [results presented by Da Prato](https://books.google.de/books?id=bxkmAwAAQBAJ&pg=PA196&lpg=PA196&dq=%22To+cover+this+important+case+we+will+introduce+the+following+assumptions%22&source=bl&ots=-3foObTT8x&sig=F7scdbxmheJDYyahqELSYcvnGP0&hl=de&sa=X&ved=0ahUKEwi0jsrApeLOAhUGUBQKHX1nDecQ6AEIIDAA#v=onepage&q=%22To%20cover%20this%20important%20case%20we%20will%20introduce%20the%20following%20assumptions%22&f=false)). $(2)$ is understood in the Stratonovich sense. Since I'm not aware of any textbook which establishes existence theory for Stratonovich SDEs, I suppose that we need to convert $(2)$ into its equivalent Itō equation $${\rm d}u(t,X\_t)=\left[\frac{\partial u}{\partial t}(t,X\_t)+\underbrace{{\rm D}u(t,X\_t)u(t,X\_t)}\_{=:\:C\_1(t,X\_t)}+\frac12\underbrace{\sum\_{n\in\mathbb N}\sqrt{\lambda\_n}\left({\rm D}\Phi(t,X\_t)\left(\Phi(t,X\_t)e\_n\right)\right)e\_n}\_{=:\:C\_2(t,X\_t)}\right]{\rm d}t+\Phi(t,X\_t){\rm d}W\_t\tag 4$$ for all $t\ge 0$, where $(e\_n)\_{n\in\mathbb N}$ is an orthonormal basis of $U$ with $$Qe\_n=\lambda\_ne\_n\;\;\;\text{for all }n\in\mathbb N\tag 5$$ for some $(\lambda\_n)\_{n\in\mathbb N}\subseteq[0,\infty)$.



> 
> My problem is that I'm unsure whether the nonlinear part of the drift in $(4)$ satisfies the usual growth and Lipschitz conditions as, for example, [presented in Da Prato's book in (7.25) and (7.26)](https://books.google.de/books?id=bxkmAwAAQBAJ&pg=PA196&lpg=PA196&dq=%22To+cover+this+important+case+we+will+introduce+the+following+assumptions%22&source=bl&ots=-3foObTT8x&sig=F7scdbxmheJDYyahqELSYcvnGP0&hl=de&sa=X&ved=0ahUKEwi0jsrApeLOAhUGUBQKHX1nDecQ6AEIIDAA#v=onepage&q=%22To%20cover%20this%20important%20case%20we%20will%20introduce%20the%20following%20assumptions%22&f=false). Namely: $$\left\|C(t,x)-C(t,y)\right\|\_H\le K\left\|x-y\right\|\_H\;\;\;\text{for all }t\ge 0\text{ and }x,y\in H\tag 6$$ and $$\left\|C(t,x)\right\|\_H\le K(1+\left\|x\right\|\_H)\;\;\;\text{for all }t\ge 0\text{ and }x\in H\tag 7$$ for some $K>0$, where $C:=C\_1+\frac12C\_2$. So, the question is: Does the drift satisfy these assumptions such that we can use the results presented in the book? And if not, can the existence of a solution be obtained in another way?
> 
> 
> 



Existence of a solution to an infinite dimensional Stratonovich SDE
In order for (2) to hold, it seems that Ito's formula needs to hold for the stochastic process $X(s)$, $0 \le s \le t$. This already requires that: (i) this process is well-defined; and (b) the Frechet partial derivatives $\partial\_t u$, $\partial\_x u$, and $\partial\_{xx} u$ are continuous and locally bounded. To check that (1) is well-defined, please write it in Ito form - which is doable since $W$ is a $Q$-Wiener process. The Ito form of (1) is then: 
$$
dX\_t = \underbrace{\left(u(t,X\_t) + \frac{1}{2} \sum\_{n \in \mathbb{N}} \sqrt{\lambda\_n} ( \partial\_x G(t,X\_t) (G(t,X\_t) e\_n)) e\_n\right)}\_{=\tilde u(t,X\_t)} dt + G(t,X\_t) dW\_t 
$$
If one insists on using existence/uniqueness theory based on Lipschitz nonlinearities, then one has to check that $\tilde u(t,x)$ is globally Lipschitz. One can do this by verifying that its Frechet partial derivative $\partial\_x \tilde u(t,x)$ is bounded uniformly in $x$ for any $t$. To obtain this bound it suffices that: for any $t>0$ and for all $x \in H$, there exists a real constant $C>0$ such that
$$
\max\left\{ \| \partial\_x u(t,x) \|\_H, \| \partial\_{xx} G(t,x) \|, \| \partial\_x G(t,x) \|, \| G(t,x) \| \right\} \le C \;.
$$ The linear growth condition follows immediately from this assumption. One can analogously obtain sufficient conditions for (4).


