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 $P_{t}D\phi$ in terms of $\mathrm{\parallel }\phi {\mathrm{\parallel }}_{L^2(H,\nu )}$, where $\nu$ is the invariant measure of $P_t$.