Hairer in his [spdes notes](http://www.hairer.org/notes/SPDEs.pdf) on pg.6, says that GFF is the stationary solution of $u\_{t}(z)=\Delta u(z)+\xi(z,t)$, where 
$\xi$ is the space-time white noise
$$\xi(x,t)=\sum \sqrt{\lambda\_{k}} B\_{k}(t)e\_{k}(x)$$
for iid Brownian motions $B\_{k}$ and L2 basis $e\_{k}$. So that means we take
$$u(x,t)=\Delta^{-1}\xi(x,t). $$
and that at each fixed time it is a GFF 
$$h(x)=\sum \frac{1}{\sqrt{\lambda\_{k}}} B\_{k}(t)e\_{k}(x).$$


In many of the standard spde textbooks, we can find existence and uniqueness of invariant measure for parabolic pdes (eg. using the Bismut-Elworthy-Li formula). But I want to know if for SHE we have stronger results eg. on the rate.



> 
> Q: Does SHE in a bounded domain with zero boundary converge to a steady state? So if $u(x,0)=GFF$ or $=0$, do we have some asymptotic results? Any rates? What is the largest function space over which it makes sense to take limits?
> 
> 
> 


In Walsh's spdes book pg.418 there is a computation for SHE on infinite domains with $d\geq 3$ showing that one obtains the Green function covariance. But is there a clean treatment for bounded domains, at least for reference purposes.


**Weak convergence**


First we show that covariances agree. For bounded domains D the formula is
$$u(x,t)=e^{t\Delta }u(x,0)+\int\_{[0,t]}\int\_{D}H(t-s,x-y)dW(s,y),$$


where the second term is a Wiener integral with Heat kernel H for domain D. 
We will compute the covariance for the bounded domain for zero initial data and $\xi(x,t):=\sum B\_{k}(t) e\_{k}(x)$. We have
$$u(x,t)=\int\_{[0,t]}\int\_{D}H(t-s,x-y)dW(s,y)$$


$$=\sum\_{k\geq 1}\int\_{0}^{t}e^{-\lambda\_{k}(t-s)}dB\_{k}(s) e\_{k}(x).$$


Therefore, by Ito isometry we indeed obtain the Green function:


$$E[u(t,x)u(t,y)]=\frac{1}{2}\sum\_{k\geq 1}\frac{e\_{k}(x)e\_{k}(y)}{\lambda\_{k}}(1-e^{-\lambda\_{k}t})\to G(x,y).$$


**Function space weak limit**
We have that the SHE $u\in C^{-\varepsilon}(\mathbb{R}\_{+},D)$ and the GFF $h\in H^{-\varepsilon}(D)$ for all $\varepsilon>0$. By Morrey's inequality
$$ H^{2}(D)\subset C^{0,\varepsilon}(D)=C^{\varepsilon}(D), $$
where $H^{2}(D)$ is the Sobolev space where second weak derivatives are also square integrable and so we also have $H^{2}(D)\subset H^{\varepsilon}(D)$ for $\varepsilon\leq 2$. So we will work with functions $f\in H^{2}(D)$.


By the covariance computation above we also obtain
$$\left \langle u(\cdot,t),f \right\rangle\_{H^{2}} \stackrel{law}{\to} \left \langle h(\cdot),f \right\rangle\_{H^{2}}.$$ 



Gaussian free field limiting distribution of additive Stochastic heat eqn bounded domain
Yes, indeed, one nice reference is ["An introduction to singular stochastic PDEs: Allen-Cahn equations, metastability and regularity structures"](https://arxiv.org/abs/1901.07420) section 2.5.1 Gaussian free field.



> 
> The stochastic heat equation (2.3.1) does not admit an invariant measure, since we have seen that its zeroth Fourier mode performs a Brownian
> motion. This problem, however, is easily cured by considering the equation
> $$\partial\_{t}\phi(t, x) = \Delta \phi(t, x) − \phi(t, x) +2 \sqrt{\epsilon}\xi(t, x) , (2.5.1)$$
> where we have reintroduced the parameter $\epsilon$ to keep track of its effect.
> 
> 
> 


Then the Gaussian measure $\mu\_{\epsilon,GFF}$ of $\sqrt{\epsilon}\phi\_{GFF}$ i.e. the GFF with covariance $\epsilon(-\Delta+1)^{-1}$ is invariant for the (2.5.1) i.e.


$$E\_{\mu\_{\epsilon,GFF}}[f(\phi(t,\cdot))]=E\_{\mu\_{\epsilon,GFF}}[f(\sqrt{\epsilon}\phi\_{GFF}(\cdot))]$$


for any integrable functional $f$.


