# Gaussian free field stationary solution additive Stochastic heat eqn

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

So is that what he meant?

• Rather than make such a drastic edit (which invalidates the posted answer) I would recommend you ask a new question. – j.c. Jun 26 '18 at 4:55
• I felt the new question was a bit too close to this one, but I will do it. – OOESCoupling Jun 26 '18 at 16:05