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

We prove an integral inequality for the invariant measure $\nu$ 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 $\nu$ in any direction $z\in H$ and that it is given by $v_z=⟨D\log \rho ,z⟩$, where $\rho$ is the density of $\nu$ with respect to the Lebesgue measure. Moreover, we prove that $v_z\in L^p(H,\nu )$ for any $p\in [1,\mathrm{\infty })$. Also we study some properties of the gradient operator in $L^p(H,\nu )$ and of his adjoint.