Hairer in his spdes notes 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 $$\left \langle u(\cdot,t),f \right\rangle_{H^{2}} \stackrel{law}{\to} \left \langle h(\cdot),f \right\rangle_{H^{2}}.$$