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

2015/12/19 by Giuseppe Da Prato, Da Prato, Giuseppe
Mathematics · #FOS: Mathematics #Probability (math.PR) #math.PR

paper · pdf · doi:10.48550/arxiv.1512.06207

arxiv created 2015/12/19 · arxiv updated 2015/12/22

Abstract

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∈ H and that it is given by v_z=⟨ Dlogρ,z⟩, where ρ is the density of ν with respect to the Lebesgue measure. Moreover, we prove that v_z∈ L^p(H,ν) for any p∈[1,∞). Also we study some properties of the gradient operator in L^p(H,ν) and of his adjoint.

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