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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

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 & 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.  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), 583-587.  Google Scholar

[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.  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), 31-55. 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), 1248-1258. 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, 229. Cambridge University Press, Cambridge, 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), 183-188.  Google Scholar

[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). Google Scholar

[9]

N. V. Krylov, Introduction to the Theory of Diffusion Processes, Translations of Mathematical Monographs, 142, 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), 396-424. doi: 10.1016/j.jfa.2005.02.001.  Google Scholar

[12]

D. Nualart, The Malliavin Calculus and Related Topics, Probability and its Applications, Springer-Verlag, 1995. doi: 10.1007/978-1-4757-2437-0.  Google Scholar

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.  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), 583-587.  Google Scholar

[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.  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), 31-55. 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), 1248-1258. 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, 229. Cambridge University Press, Cambridge, 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), 183-188.  Google Scholar

[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). Google Scholar

[9]

N. V. Krylov, Introduction to the Theory of Diffusion Processes, Translations of Mathematical Monographs, 142, 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), 396-424. doi: 10.1016/j.jfa.2005.02.001.  Google Scholar

[12]

D. Nualart, The Malliavin Calculus and Related Topics, Probability and its Applications, Springer-Verlag, 1995. doi: 10.1007/978-1-4757-2437-0.  Google Scholar

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