Gaussian free field limiting distribution of additive Stochastic heat eqn bounded domain

Question

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

Answer 1

Yes, indeed, one nice reference is "An introduction to singular stochastic PDEs: Allen-Cahn equations, metastability and regularity structures" 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 $f$.