Estimate for $P_tD$ for the stochastic Burgers equation

TL;DR

This paper derives a new formula for the derivative of the transition semigroup associated with the stochastic Burgers equation, leading to novel insights into the invariant measure's properties, including Fomin differentiability and an integration by parts formula.

Contribution

It introduces a new formula for $P_tDoldsymbol{ extphi}$ that depends only on $oldsymbol{ extphi}$, not its derivatives, advancing understanding of the stochastic Burgers equation's invariant measure.

Findings

New formula for $P_tDoldsymbol{ extphi}$ independent of $oldsymbol{ extphi}$'s derivatives

Results on Fomin differentiability of the invariant measure

Generalized integration by parts formula for the invariant measure

Abstract

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 $P_tD\varphi$ (where $\varphi:H\to\R$ is bounded and Borel) which depends on $\varphi$ 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.