Estimate for $P_tD$ 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 $P_tDφ$ (where $φ:H\to\R$ is bounded and Borel) which depends on $φ$ but not on its derivative. Then we deduce some new consequences for the invariant measure $ν$ of $P_t$ as its Fomin differentiability and an integration by parts formula which generalises the classical one for gaussian measures.

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