American Institute of Mathematical Sciences

Advanced
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.  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

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.  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

[1]

Zhongkai Guo. Invariant foliations for stochastic partial differential equations with dynamic boundary conditions. Discrete & Continuous Dynamical Systems - A, 2015, 35 (11) : 5203-5219. doi: 10.3934/dcds.2015.35.5203
[2]

Yan Wang, Guanggan Chen. Invariant measure of stochastic fractional Burgers equation with degenerate noise on a bounded interval. Communications on Pure & Applied Analysis, 2019, 18 (6) : 3121-3135. doi: 10.3934/cpaa.2019140
[3]

Shouchuan Hu, Nikolaos S. Papageorgiou. Nonlinear Neumann equations driven by a nonhomogeneous differential operator. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1055-1078. doi: 10.3934/cpaa.2011.10.1055
[4]

András Bátkai, Istvan Z. Kiss, Eszter Sikolya, Péter L. Simon. Differential equation approximations of stochastic network processes: An operator semigroup approach. Networks & Heterogeneous Media, 2012, 7 (1) : 43-58. doi: 10.3934/nhm.2012.7.43
[5]

Bernd Aulbach, Martin Rasmussen, Stefan Siegmund. Invariant manifolds as pullback attractors of nonautonomous differential equations. Discrete & Continuous Dynamical Systems - A, 2006, 15 (2) : 579-596. doi: 10.3934/dcds.2006.15.579
[6]

Ludwig Arnold, Igor Chueshov. Cooperative random and stochastic differential equations. Discrete & Continuous Dynamical Systems - A, 2001, 7 (1) : 1-33. doi: 10.3934/dcds.2001.7.1
[7]

Zhiyuan Wen, Meirong Zhang. On the optimization problems of the principal eigenvalues of measure differential equations with indefinite measures. Discrete & Continuous Dynamical Systems - B, 2020, 25 (8) : 3257-3274. doi: 10.3934/dcdsb.2020061
[8]

Hua Liu, Zhaosheng Feng. Begehr-Hile operator and its applications to some differential equations. Communications on Pure & Applied Analysis, 2010, 9 (2) : 387-395. doi: 10.3934/cpaa.2010.9.387
[9]

Rongmei Cao, Jiangong You. The existence of integrable invariant manifolds of Hamiltonian partial differential equations. Discrete & Continuous Dynamical Systems - A, 2006, 16 (1) : 227-234. doi: 10.3934/dcds.2006.16.227
[10]

Can Huang, Zhimin Zhang. The spectral collocation method for stochastic differential equations. Discrete & Continuous Dynamical Systems - B, 2013, 18 (3) : 667-679. doi: 10.3934/dcdsb.2013.18.667
[11]

Jasmina Djordjević, Svetlana Janković. Reflected backward stochastic differential equations with perturbations. Discrete & Continuous Dynamical Systems - A, 2018, 38 (4) : 1833-1848. doi: 10.3934/dcds.2018075
[12]

Arnulf Jentzen. Taylor expansions of solutions of stochastic partial differential equations. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 515-557. doi: 10.3934/dcdsb.2010.14.515
[13]

Jan A. Van Casteren. On backward stochastic differential equations in infinite dimensions. Discrete & Continuous Dynamical Systems - S, 2013, 6 (3) : 803-824. doi: 10.3934/dcdss.2013.6.803
[14]

Igor Chueshov, Michael Scheutzow. Invariance and monotonicity for stochastic delay differential equations. Discrete & Continuous Dynamical Systems - B, 2013, 18 (6) : 1533-1554. doi: 10.3934/dcdsb.2013.18.1533
[15]

A. Alamo, J. M. Sanz-Serna. Word combinatorics for stochastic differential equations: Splitting integrators. Communications on Pure & Applied Analysis, 2019, 18 (4) : 2163-2195. doi: 10.3934/cpaa.2019097
[16]

Pingping Niu, Shuai Lu, Jin Cheng. On periodic parameter identification in stochastic differential equations. Inverse Problems & Imaging, 2019, 13 (3) : 513-543. doi: 10.3934/ipi.2019025
[17]

Yaozhong Hu, David Nualart, Xiaobin Sun, Yingchao Xie. Smoothness of density for stochastic differential equations with Markovian switching. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 3615-3631. doi: 10.3934/dcdsb.2018307
[18]

Yongqiang Suo, Chenggui Yuan. Large deviations for neutral stochastic functional differential equations. Communications on Pure & Applied Analysis, 2020, 19 (4) : 2369-2384. doi: 10.3934/cpaa.2020103
[19]

Lorenzo Zambotti. A brief and personal history of stochastic partial differential equations. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020264
[20]

Xin Chen, Ana Bela Cruzeiro. Stochastic geodesics and forward-backward stochastic differential equations on Lie groups. Conference Publications, 2013, 2013 (special) : 115-121. doi: 10.3934/proc.2013.2013.115

2019 Impact Factor: 1.27

Tools

Metrics

  • PDF downloads (37)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences