An integral inequality for the invariant measure of some finite dimensional stochastic differential equation
Giuseppe Da Prato · Discrete and Continuous Dynamical Systems - B · 2016
We prove an integral inequality for the invariant measure $ u$ of a stochastic differential equation with additive noise in a finite dimensional space $H=\mathbb R^d$. As a consequence, we show that there exists the Fomin derivative of $ u$ in any direction $z\in H$ and that it is given by $v_z=\langle D\log\rho,z\rangle$, where $\rho$ is the density of $ u$ with respect to the Lebesgue measure. Moreover, we prove that $v_z\in L^p(H, u)$ for any $p\in[1,\infty)$. Also we study some properties of the gradient operator in $L^p(H, u)$ and of his adjoint.