# American Institute of Mathematical Sciences

November  2016, 21(9): 3015-3027. doi: 10.3934/dcdsb.2016085

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

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

Received  November 2015 Revised  February 2016 Published  October 2016

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.
Citation: Giuseppe Da Prato. An integral inequality for the invariant measure of some finite dimensional stochastic differential equation. Discrete & Continuous Dynamical Systems - B, 2016, 21 (9) : 3015-3027. doi: 10.3934/dcdsb.2016085
##### References:
 [1] V. I. Bogachev, N. V. Krylov and M. Röckner, On regularity of transition probabilities and invariant measures of singular diffusions under minimal conditions,, Comm. Partial Differential Equations, 26 (2001), 2037.  doi: 10.1081/PDE-100107815.  Google Scholar [2] V. I. Bogachev, N. V. Krylov and M. Röckner, Regularity and global bounds for the densities of invariant measures of diffusion processes,, (Russian) Dokl. Akad. Nauk, 405 (2005), 583.   Google Scholar [3] G. Da Prato and A. Debussche, Ergodicity for the $3D$ stochastic Navier-Stokes equations,, J. Math. Pures Appl., 82 (2003), 877.  doi: 10.1016/S0021-7824(03)00025-4.  Google Scholar [4] G. Da Prato and A. Debussche, $m$-dissipativity of Kolmogorov operators corresponding to Burgers equations with space-time white noise,, Potential Anal, 26 (2007), 31.  doi: 10.1007/s11118-006-9021-5.  Google Scholar [5] G. Da Prato and A. Debussche, Estimate for $P_tD$ for the stochastic Burgers equation,, Ann. Inst. Henri Poincaré Probab. Stat., 52 (2016), 1248.  doi: 10.1214/15-AIHP685.  Google Scholar [6] G. Da Prato and J. Zabczyk, Ergodicity for Infinite-Dimensional Systems,, London Mathematical Society Lecture Note Series, (1996).  doi: 10.1017/CBO9780511662829.  Google Scholar [7] G. Da Prato and J. Zabczyk, Differentiability of the Feynman-Kac semigroup and a control application,, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl., 8 (1997), 183.   Google Scholar [8] K. D. Elworthy, Stochastic flows on Riemannian manifolds,, in Diffusion processes and related problems in analysis, (1992), 33.   Google Scholar [9] N. V. Krylov, Introduction to the Theory of Diffusion Processes,, Translations of Mathematical Monographs, (1995).   Google Scholar [10] P. Malliavin, Stochastic Analysis,, Springer-Verlag, (1997).  doi: 10.1007/978-3-642-15074-6.  Google Scholar [11] G. Metafune, D. Pallara and A. Rhandi, Global properties of invariant measures,, J. Funct. Analysis, 223 (2005), 396.  doi: 10.1016/j.jfa.2005.02.001.  Google Scholar [12] D. Nualart, The Malliavin Calculus and Related Topics,, Probability and its Applications, (1995).  doi: 10.1007/978-1-4757-2437-0.  Google Scholar

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##### References:
 [1] V. I. Bogachev, N. V. Krylov and M. Röckner, On regularity of transition probabilities and invariant measures of singular diffusions under minimal conditions,, Comm. Partial Differential Equations, 26 (2001), 2037.  doi: 10.1081/PDE-100107815.  Google Scholar [2] V. I. Bogachev, N. V. Krylov and M. Röckner, Regularity and global bounds for the densities of invariant measures of diffusion processes,, (Russian) Dokl. Akad. Nauk, 405 (2005), 583.   Google Scholar [3] G. Da Prato and A. Debussche, Ergodicity for the $3D$ stochastic Navier-Stokes equations,, J. Math. Pures Appl., 82 (2003), 877.  doi: 10.1016/S0021-7824(03)00025-4.  Google Scholar [4] G. Da Prato and A. Debussche, $m$-dissipativity of Kolmogorov operators corresponding to Burgers equations with space-time white noise,, Potential Anal, 26 (2007), 31.  doi: 10.1007/s11118-006-9021-5.  Google Scholar [5] G. Da Prato and A. Debussche, Estimate for $P_tD$ for the stochastic Burgers equation,, Ann. Inst. Henri Poincaré Probab. Stat., 52 (2016), 1248.  doi: 10.1214/15-AIHP685.  Google Scholar [6] G. Da Prato and J. Zabczyk, Ergodicity for Infinite-Dimensional Systems,, London Mathematical Society Lecture Note Series, (1996).  doi: 10.1017/CBO9780511662829.  Google Scholar [7] G. Da Prato and J. Zabczyk, Differentiability of the Feynman-Kac semigroup and a control application,, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl., 8 (1997), 183.   Google Scholar [8] K. D. Elworthy, Stochastic flows on Riemannian manifolds,, in Diffusion processes and related problems in analysis, (1992), 33.   Google Scholar [9] N. V. Krylov, Introduction to the Theory of Diffusion Processes,, Translations of Mathematical Monographs, (1995).   Google Scholar [10] P. Malliavin, Stochastic Analysis,, Springer-Verlag, (1997).  doi: 10.1007/978-3-642-15074-6.  Google Scholar [11] G. Metafune, D. Pallara and A. Rhandi, Global properties of invariant measures,, J. Funct. Analysis, 223 (2005), 396.  doi: 10.1016/j.jfa.2005.02.001.  Google Scholar [12] D. Nualart, The Malliavin Calculus and Related Topics,, Probability and its Applications, (1995).  doi: 10.1007/978-1-4757-2437-0.  Google Scholar
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