Hairer in his [spdes notes][1] 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 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?

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 $f\in H^{2}(D)$.

Now I am trying to see if we can modify [theorem 2.3][2] to obtain 
$$\left \langle u(\cdot,t),f \right\rangle_{H^{2}} \to \left \langle h(\cdot),f \right\rangle_{H^{2}}.$$
That theorem was proved for tempered distributions over $\mathbb{R}^{d}$ (not bounded domains with good enough boundaries). 

>Q: Any clean results for checking $u\stackrel{weakly}{\to} h$ i.e. 
$$\left \langle u(\cdot,t),f \right\rangle_{H} \to \left \langle h(\cdot),f \right\rangle_{H}?$$ 


At the moment I am trying to use invariant measure existence and uniqueness results to prove that GFF is the limiting distribution. 





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][3] 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. 


  [1]: http://www.hairer.org/notes/SPDEs.pdf
  [2]: https://arxiv.org/abs/1706.09326
  [3]: https://arxiv.org/abs/1609.08447