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

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]

Fabio Camilli, Giulia Cavagnari, Raul De Maio, Benedetto Piccoli. Superposition principle and schemes for measure differential equations. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2020050

[2]

Lorenzo Zambotti. A brief and personal history of stochastic partial differential equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 471-487. doi: 10.3934/dcds.2020264

[3]

Yueyang Zheng, Jingtao Shi. A stackelberg game of backward stochastic differential equations with partial information. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020047

[4]

Fathalla A. Rihan, Hebatallah J. Alsakaji. Stochastic delay differential equations of three-species prey-predator system with cooperation among prey species. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020468

[5]

Martin Heida, Stefan Neukamm, Mario Varga. Stochastic homogenization of $ \Lambda $-convex gradient flows. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 427-453. doi: 10.3934/dcdss.2020328

[6]

Siyang Cai, Yongmei Cai, Xuerong Mao. A stochastic differential equation SIS epidemic model with regime switching. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020317

[7]

Stefan Ruschel, Serhiy Yanchuk. The spectrum of delay differential equations with multiple hierarchical large delays. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 151-175. doi: 10.3934/dcdss.2020321

[8]

Zhenzhen Wang, Tianshou Zhou. Asymptotic behaviors and stochastic traveling waves in stochastic Fisher-KPP equations. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020323

[9]

Thabet Abdeljawad, Mohammad Esmael Samei. Applying quantum calculus for the existence of solution of $ q $-integro-differential equations with three criteria. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020440

[10]

Xuhui Peng, Rangrang Zhang. Approximations of stochastic 3D tamed Navier-Stokes equations. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5337-5365. doi: 10.3934/cpaa.2020241

[11]

Christian Beck, Lukas Gonon, Martin Hutzenthaler, Arnulf Jentzen. On existence and uniqueness properties for solutions of stochastic fixed point equations. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020320

[12]

Min Xi, Wenyu Sun, Jun Chen. Survey of derivative-free optimization. Numerical Algebra, Control & Optimization, 2020, 10 (4) : 537-555. doi: 10.3934/naco.2020050

[13]

Mostafa Mbekhta. Representation and approximation of the polar factor of an operator on a Hilbert space. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020463

[14]

Yangrong Li, Shuang Yang, Qiangheng Zhang. Odd random attractors for stochastic non-autonomous Kuramoto-Sivashinsky equations without dissipation. Electronic Research Archive, 2020, 28 (4) : 1529-1544. doi: 10.3934/era.2020080

[15]

Pengyu Chen. Non-autonomous stochastic evolution equations with nonlinear noise and nonlocal conditions governed by noncompact evolution families. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020383

[16]

Lin Shi, Xuemin Wang, Dingshi Li. Limiting behavior of non-autonomous stochastic reaction-diffusion equations with colored noise on unbounded thin domains. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5367-5386. doi: 10.3934/cpaa.2020242

[17]

Predrag S. Stanimirović, Branislav Ivanov, Haifeng Ma, Dijana Mosić. A survey of gradient methods for solving nonlinear optimization. Electronic Research Archive, 2020, 28 (4) : 1573-1624. doi: 10.3934/era.2020115

[18]

Awais Younus, Zoubia Dastgeer, Nudrat Ishaq, Abdul Ghaffar, Kottakkaran Sooppy Nisar, Devendra Kumar. On the observability of conformable linear time-invariant control systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020444

[19]

Fanni M. Sélley. A self-consistent dynamical system with multiple absolutely continuous invariant measures. Journal of Computational Dynamics, 2021, 8 (1) : 9-32. doi: 10.3934/jcd.2021002

[20]

Wenqiang Zhao, Yijin Zhang. High-order Wong-Zakai approximations for non-autonomous stochastic $ p $-Laplacian equations on $ \mathbb{R}^N $. Communications on Pure & Applied Analysis, 2021, 20 (1) : 243-280. doi: 10.3934/cpaa.2020265

2019 Impact Factor: 1.27

Metrics

  • PDF downloads (45)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]