Existence of the Fomin derivative of the invariant measure of a stochastic reaction--diffusion equation

We consider a reaction--diffusion equation perturbed by noise (not necessarily white). We prove existence of the Fomin derivative of the corresponding transition semigroup $P_t$. The main tool is a new estimate for $P_tD\varphi$ in terms of $\|\varphi\|_{L^2(H,ν)}$, where $ν$ is the invariant measure of $P_t$.