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

TL;DR

This paper establishes an integral inequality for the invariant measure of certain finite-dimensional stochastic differential equations, leading to new insights into the measure's derivatives and integrability properties.

Contribution

It introduces a novel integral inequality for the invariant measure and characterizes its Fomin derivative, advancing understanding of measure regularity in stochastic differential equations.

Findings

Existence of the Fomin derivative of the invariant measure in any direction.

Representation of the derivative as the inner product with the gradient of the log-density.

The derivative belongs to L^p spaces for all p in [1, ∞).

Abstract

We prove an integral inequality for the invariant measure $\nu$ 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 $\nu$ 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 $\nu$ with respect to the Lebesgue measure. Moreover, we prove that $v_z\in L^p(H,\nu)$ for any $p\in[1,\infty)$. Also we study some properties of the gradient operator in $L^p(H,\nu)$ and of his adjoint.