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 $$h(x)=\sum \frac{1}{\sqrt{\lambda_{k}}} B_{k}(t)e_{k}(x).$$

Q: Does SHE in a bounded domain converge to a steady state? So if $u(x,0)=GFF$ or $=0$, do we have some asymptotic results? Any rates?

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.

For the infinite domain, from the same notes, if we compute and expand the covariance for SHE we have

$$E(u(t,x)u(t,y))=log(\frac{1}{|x-y|^{2}})+log(t)+c+O(\frac{|x-y|^{2}}{t})$$

and so even though for each fixed we have a GFF like object, for $t\to \infty$ the covariance becomes infinite. This is fine because for the whole plane we don't even have the usual GFF (meaning the one whose covariance is the Green function but what is called the "Whole-plane GFF").

The closest thing I found in the literature is about stochastic quantization and a special case from there gives that $$u_{t}=\Delta u-u+\xi(x,t)$$ has the GFF as the limiting distribution.