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

Giuseppe Da  Prato

doi:10.3934/dcdsb.2016085

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2016, Volume 21, Issue 9: 3015-3027. Doi: 10.3934/dcdsb.2016085

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

Giuseppe Da Prato^1,

1.
Scuola Normale Superiore di Pisa, Piazza dei Cavalieri 7, 56126 Pisa

Received: November 2015

Revised: February 2016

Published: November 2016

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Abstract

We prove an integral inequality for the invariant measure $\nu$ of a stochastic differential equation with additive noise in a finite dimensional space $H=\mathbb R^d$. 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 $v_z=\langle D\log\rho,z\rangle$, 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.

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Mathematics Subject Classification: 60H07, 60H30, 37L40.

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References

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