We know that the solution of a linear SDE is a Gaussian process. I wonder whether the same is true for a SPDE casted to an infinite-dimensional SDE on a Hilbert space $H$ like $${\rm d}X_t=f(t)AX_t+\sigma W_t\tag1,$$ where $A$ is a (possibly unbounded) operator on $H$, $f:[0,T]\to\mathbb R$ and $W$ is a cylindrical Wiener process on $H$. The example that I've god in mind is $H=L^2(D)$, where $D\subseteq\mathbb R^d$ is bounded and open, and $A=\Delta u$ on $H^2(D)$. Assuming $X_0$ is deterministic or Gaussian, can we show that $X$ is then a Gaussian process? (In the sense of the Gaussian distribution on Hilbert spaces: http://www.kevinoconnor.co/wp-content/uploads/2018/05/GaussianMeasuresOnHilbertSpaces.pdf)?
We cannot simply mimic the proof from finite-dimensions, since here $A$ is not defined on all of $H$ ...