Estimate for $P_{t}D$ for the stochastic Burgers equation

We consider the Burgers equation on $H=L^2(0,1)$ perturbed by white noise and the corresponding transition semigroup $P_t$. We prove a new formula for $P_{t}D\phi$ (where $\phi :H\to \mathbb{R}$ is bounded and Borel) which depends on $\phi$ but not on its derivative. Then we deduce some new consequences for the invariant measure $\nu$ of $P_t$ as its Fomin differentiability and an integration by parts formula which generalises the classical one for gaussian measures.