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

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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

Giuseppe Da Prato ^1,

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

Received November 2015 Revised February 2016 Published October 2016

Abstract
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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.

Keywords: Stochastic differential equations, Fomin derivative, invariant measure, gradient operator..

Mathematics Subject Classification: 60H07, 60H30, 37L4.

Citation: Giuseppe Da Prato. An integral inequality for the invariant measure of some finite dimensional stochastic differential equation. Discrete and 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), 2037-2080. doi: 10.1081/PDE-100107815.
[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), 583-587.
[3]	G. Da Prato and A. Debussche, Ergodicity for the $3D$ stochastic Navier-Stokes equations, J. Math. Pures Appl., 82 (2003), 877-947. doi: 10.1016/S0021-7824(03)00025-4.
[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), 31-55. doi: 10.1007/s11118-006-9021-5.
[5]	G. Da Prato and A. Debussche, Estimate for $P_tD$ for the stochastic Burgers equation, Ann. Inst. Henri Poincaré Probab. Stat., 52 (2016), 1248-1258. doi: 10.1214/15-AIHP685.
[6]	G. Da Prato and J. Zabczyk, Ergodicity for Infinite-Dimensional Systems, London Mathematical Society Lecture Note Series, 229. Cambridge University Press, Cambridge, 1996. doi: 10.1017/CBO9780511662829.
[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), 183-188.
[8]	K. D. Elworthy, Stochastic flows on Riemannian manifolds, in Diffusion processes and related problems in analysis, Vol. II, M. A. Pinsky and V. Wihstutz (eds.), Birkhäuser, 33-72, 1992. MR1187985 Reviewed Elworthy, K. D. Stochastic flows on Riemannian manifolds. Diffusion processes and related problems in analysis, Vol. II (Charlotte, NC, 1990), 37-72, Progr. Probab., 27, Birkhäuser Boston, Boston, MA, 1992. 58G32 (60H10).
[9]	N. V. Krylov, Introduction to the Theory of Diffusion Processes, Translations of Mathematical Monographs, 142, 1995.
[10]	P. Malliavin, Stochastic Analysis, Springer-Verlag, 1997. doi: 10.1007/978-3-642-15074-6.
[11]	G. Metafune, D. Pallara and A. Rhandi, Global properties of invariant measures, J. Funct. Analysis, 223 (2005), 396-424. doi: 10.1016/j.jfa.2005.02.001.
[12]	D. Nualart, The Malliavin Calculus and Related Topics, Probability and its Applications, Springer-Verlag, 1995. doi: 10.1007/978-1-4757-2437-0.

show all references

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), 2037-2080. doi: 10.1081/PDE-100107815.
[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), 583-587.
[3]	G. Da Prato and A. Debussche, Ergodicity for the $3D$ stochastic Navier-Stokes equations, J. Math. Pures Appl., 82 (2003), 877-947. doi: 10.1016/S0021-7824(03)00025-4.
[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), 31-55. doi: 10.1007/s11118-006-9021-5.
[5]	G. Da Prato and A. Debussche, Estimate for $P_tD$ for the stochastic Burgers equation, Ann. Inst. Henri Poincaré Probab. Stat., 52 (2016), 1248-1258. doi: 10.1214/15-AIHP685.
[6]	G. Da Prato and J. Zabczyk, Ergodicity for Infinite-Dimensional Systems, London Mathematical Society Lecture Note Series, 229. Cambridge University Press, Cambridge, 1996. doi: 10.1017/CBO9780511662829.
[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), 183-188.
[8]	K. D. Elworthy, Stochastic flows on Riemannian manifolds, in Diffusion processes and related problems in analysis, Vol. II, M. A. Pinsky and V. Wihstutz (eds.), Birkhäuser, 33-72, 1992. MR1187985 Reviewed Elworthy, K. D. Stochastic flows on Riemannian manifolds. Diffusion processes and related problems in analysis, Vol. II (Charlotte, NC, 1990), 37-72, Progr. Probab., 27, Birkhäuser Boston, Boston, MA, 1992. 58G32 (60H10).
[9]	N. V. Krylov, Introduction to the Theory of Diffusion Processes, Translations of Mathematical Monographs, 142, 1995.
[10]	P. Malliavin, Stochastic Analysis, Springer-Verlag, 1997. doi: 10.1007/978-3-642-15074-6.
[11]	G. Metafune, D. Pallara and A. Rhandi, Global properties of invariant measures, J. Funct. Analysis, 223 (2005), 396-424. doi: 10.1016/j.jfa.2005.02.001.
[12]	D. Nualart, The Malliavin Calculus and Related Topics, Probability and its Applications, Springer-Verlag, 1995. doi: 10.1007/978-1-4757-2437-0.

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