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An integral inequality for the invariant measure of some finite dimensional stochastic differential equation

We prove an integral inequality for the invariant measure $ν$ of a stochastic differential equation with additive noise in a finite dimensional space $H=\R^d$. As a consequence, we show that there exists the Fomin derivative of $ν$ in any direction $z\in H$ and that it is given by $v_z=\langle D\logρ,z\rangle$, where $ρ$ is the density of $ν$ with respect to the Lebesgue measure. Moreover, we prove that $v_z\in L^p(H,ν)$ for any $p\in[1,\infty)$. Also we study some properties of the gradient operator in $L^p(H,ν)$ and of his adjoint.