Extended Poisson equation for weakly ergodic Markov processes
Authors:
A. Yu. Veretennikov and A. M. Kulik
Translated by:
N. Semenov
Journal:
Theor. Probability and Math. Statist. 85 (2012), 23-39
MSC (2010):
Primary 60H10, 60J10
DOI:
https://doi.org/10.1090/S0094-9000-2013-00871-0
Published electronically:
January 11, 2013
Full-text PDF Free Access
Abstract |
References |
Similar Articles |
Additional Information
Abstract: Solvability conditions for a Poisson equation with an extended generator of a general Markov process are obtained. The predictable part in the Doob–Meyer decomposition is described for a process of the form $g(X(t), Y(t))$, where $Y$ is a solution of a stochastic equation with the coefficients depending on $X$ and where the function $g=g(x,y)$ is defined as a family of solutions of the Poisson equation.
- N. Abourashchi and A. Yu. Veretennikov, On stochastic averaging and mixing, Theory Stoch. Process. 16 (2010), no. 1, 111–129. MR 2779833
- R. Z. Has′minskiĭ, A limit theorem for solutions of differential equations with a random right hand part, Teor. Verojatnost. i Primenen 11 (1966), 444–462 (Russian, with English summary). MR 0203789
- A. N. Borodin, A limit theorem for the solutions of differential equations with a random right-hand side, Teor. Verojatnost. i Primenen. 22 (1977), no. 3, 498–512 (Russian, with English summary). MR 0517995
- V. V. Sarafyan and A. V. Skorokhod, Dynamical systems with fast switchings, Teor. Veroyatnost. i Primenen. 32 (1987), no. 4, 658–669 (Russian). MR 927247
- Stewart N. Ethier and Thomas G. Kurtz, Markov processes, Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics, John Wiley & Sons, Inc., New York, 1986. Characterization and convergence. MR 838085
- Vladimir S. Koroliuk and Nikolaos Limnios, Stochastic systems in merging phase space, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2005. MR 2205562
- E. Pardoux and A. Yu. Veretennikov, On the Poisson equation and diffusion approximation. I, Ann. Probab. 29 (2001), no. 3, 1061–1085. MR 1872736, DOI https://doi.org/10.1214/aop/1015345596
- A. Yu. Veretennikov, Estimates of the mixing rate for stochastic equations, Teor. Veroyatnost. i Primenen. 32 (1987), no. 2, 299–308 (Russian). MR 902757
- A. Yu. Veretennikov, On polynomial mixing bounds for stochastic differential equations, Stochastic Process. Appl. 70 (1997), no. 1, 115–127. MR 1472961, DOI https://doi.org/10.1016/S0304-4149%2897%2900056-2
- S. A. Klokov and A. Yu. Veretennikov, Sub-exponential mixing rate for a class of Markov chains, Math. Commun. 9 (2004), no. 1, 9–26. MR 2076227
- D. Down, S. P. Meyn, and R. L. Tweedie, Exponential and uniform ergodicity of Markov processes, Ann. Probab. 23 (1995), no. 4, 1671–1691. MR 1379163
- Randal Douc, Gersende Fort, and Arnaud Guillin, Subgeometric rates of convergence of $f$-ergodic strong Markov processes, Stochastic Process. Appl. 119 (2009), no. 3, 897–923. MR 2499863, DOI https://doi.org/10.1016/j.spa.2008.03.007
- Alexey M. Kulik, Exponential ergodicity of the solutions to SDE’s with a jump noise, Stochastic Process. Appl. 119 (2009), no. 2, 602–632. MR 2494006, DOI https://doi.org/10.1016/j.spa.2008.02.006
- Giuseppe Da Prato and Jerzy Zabczyk, Stochastic equations in infinite dimensions, Encyclopedia of Mathematics and its Applications, vol. 44, Cambridge University Press, Cambridge, 1992. MR 1207136
- S. Peszat and J. Zabczyk, Stochastic partial differential equations with Lévy noise, Encyclopedia of Mathematics and its Applications, vol. 113, Cambridge University Press, Cambridge, 2007. An evolution equation approach. MR 2356959
- Martin Hairer and Jonathan C. Mattingly, Ergodicity of the 2D Navier-Stokes equations with degenerate stochastic forcing, Ann. of Math. (2) 164 (2006), no. 3, 993–1032. MR 2259251, DOI https://doi.org/10.4007/annals.2006.164.993
- Martin Hairer and Jonathan C. Mattingly, Spectral gaps in Wasserstein distances and the 2D stochastic Navier-Stokes equations, Ann. Probab. 36 (2008), no. 6, 2050–2091. MR 2478676, DOI https://doi.org/10.1214/08-AOP392
- Tomasz Komorowski, Szymon Peszat, and Tomasz Szarek, On ergodicity of some Markov processes, Ann. Probab. 38 (2010), no. 4, 1401–1443. MR 2663632, DOI https://doi.org/10.1214/09-AOP513
- Tomasz Szarek, The uniqueness of invariant measures for Markov operators, Studia Math. 189 (2008), no. 3, 225–233. MR 2457488, DOI https://doi.org/10.4064/sm189-3-2
- M. Hairer, J. C. Mattingly, and M. Scheutzow, Asymptotic coupling and a general form of Harris’ theorem with applications to stochastic delay equations, Prob. Theory Rel. Fields (2009), DOI 10.1007/s00440-009-0250-6.
- È. Pardoux and A. Yu. Veretennikov, On Poisson equation and diffusion approximation. II, Ann. Probab. 31 (2003), no. 3, 1166–1192. MR 1988467, DOI https://doi.org/10.1214/aop/1055425774
- A. Yu. Veretennikov and A. M. Kulik, The extended Poisson equation for weakly ergodic Markov processes, Teor. Ĭmovīr. Mat. Stat. 85 (2011), 22–38 (Russian, with English, Russian and Ukrainian summaries). MR 2933700
- Hiroshi Kunita, Absolute continuity of Markov processes and generators, Nagoya Math. J. 36 (1969), 1–26. MR 250387
- Daniel Revuz and Marc Yor, Continuous martingales and Brownian motion, 3rd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 293, Springer-Verlag, Berlin, 1999. MR 1725357
- Zbigniew Palmowski and Tomasz Rolski, A technique for exponential change of measure for Markov processes, Bernoulli 8 (2002), no. 6, 767–785. MR 1963661
- Vladimir M. Zolotarev, Modern theory of summation of random variables, Modern Probability and Statistics, VSP, Utrecht, 1997. MR 1640024
- R. M. Dudley, Real analysis and probability, Cambridge Studies in Advanced Mathematics, vol. 74, Cambridge University Press, Cambridge, 2002. Revised reprint of the 1989 original. MR 1932358
- Nobuyuki Ikeda and Shinzo Watanabe, Stochastic differential equations and diffusion processes, North-Holland Mathematical Library, vol. 24, North-Holland Publishing Co., Amsterdam-New York; Kodansha, Ltd., Tokyo, 1981. MR 637061
- R. L. Dobrušin, Definition of a system of random variables by means of conditional distributions, Teor. Verojatnost. i Primenen. 15 (1970), 469–497 (Russian, with English summary). MR 0298716
- Ĭ. Ī. Gīhman and A. V. Skorohod, Stochastic differential equations, Springer-Verlag, New York-Heidelberg, 1972. Translated from the Russian by Kenneth Wickwire; Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 72. MR 0346904
- N. Abourashchi and A. Yu. Veretennikov, On stochastic averaging and mixing, Theory Stoch. Process. 16(32) (2010), no. 1, 111–129. MR 2779833 (2012a:60188)
- R. Z. Khas’minskiĭ, A limit theorem for the solutions of differential equations with random right-hand sides, 11 (1966), no. 3, 444–462; English transl. in Theor. Probability Appl. 11 (1966), no. 3, 390–406. MR 0203789 (34:3637)
- A. N. Borodin, A limit theorem for the solutions of differential equations with a random right-hand side, Teor. Verojatnost. i Primenen. 22 (1977), no. 3, 498–512; English transl. in Theor. Probability Appl. 22 (1977), no. 3, 482–497. MR 0517995 (58:24546)
- V. V. Sarafyan and A. V. Skorokhod, Dynamical systems with fast switchings, Teor. Verojatnost. i Primenen. 32 (1987), no. 4, 658–669; English transl. in Theory Probab. Appl. 32 (1987), no. 4, 595–607. MR 927247 (89f:60064)
- S. N. Ethier and T. G. Kurtz, Markov Processes. Characterization and Convergence, Wiley, New York, 1986. MR 838085 (88a:60130)
- V. S. Korolyuk and N. Limnios, Stochastic Systems in Merging Phase Space, World Scientific, New Jersey, 2005. MR 2205562 (2007a:60004)
- E. Pardoux and A. Yu. Veretennikov, On Poisson equation and diffusion approximation 1, Ann. Probab. 29 (2001), 1061–1085. MR 1872736 (2002j:60120)
- A. Yu. Veretennikov, Bounds for the mixing rates in the theory of stochastic equations, Theory Probab. Appl. 32 (1987), 273–281. MR 902757 (89b:60144)
- A. Yu. Veretennikov, On polynomial mixing bounds for stochastic differential equations, Stoch. Process. Appl. 70 (1997), 115–127. MR 1472961 (99k:60158)
- S. A. Klokov and A. Yu. Veretennikov, Sub-exponential mixing rate for a class of Markov chains, Math. Comm. 9 (2004), 9–26. MR 2076227 (2005f:60148)
- D. Down, S. P. Meyn, and R. L. Tweedie, Exponential and uniform ergodicity of Markov processes, Ann. Probab. 23 (1995) no. 4, 1671–1691. MR 1379163 (97c:60181)
- R. Douc, G. Fort, and A. Guillin, Subgeometric rates of convergence of $f$-ergodic strong Markov processes, Stoch. Process. Appl. 119 (2009), no. 3, 897–923. MR 2499863 (2010j:60184)
- A. M. Kulik, Exponential ergodicity of the solutions to SDE’s with a jump noise, Stoch. Proc. Appl. 119 (2009), 602–632. MR 2494006 (2010i:60176)
- G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge University Press, Cambridge, 1992. MR 1207136 (95g:60073)
- S. Peszat and J. Zabczyk, Stochastic Partial Differential Equations with Lévy Noise (an Evolution Equation Approach), Cambridge University Press, Cambridge, 2007. MR 2356959 (2009b:60200)
- M. Hairer and J. and Mattingly, Ergodicity of the 2D Navier–Stokes equations with degenerate stochastic forcing, Ann. Math. 164 (2006), 993–1032. MR 2259251 (2008a:37095)
- M. Hairer and J. Mattingly, Spectral gaps in Wasserstein distances and the 2D stochastic Navier–Stokes equations, Ann. Probab. 36 (2008), 2050–2091. MR 2478676 (2010i:35295)
- T. Komorowski, Sz. Peszat, and T. Szarek, On ergodicity of some Markov processes, Ann. Probab. 38 (2010), 1401–1443. MR 2663632 (2011f:60148)
- T. Szarek, The uniqueness of invariant measures for Markov operators, Studia Math. 189 (2008), 225–233. MR 2457488 (2009m:60166)
- M. Hairer, J. C. Mattingly, and M. Scheutzow, Asymptotic coupling and a general form of Harris’ theorem with applications to stochastic delay equations, Prob. Theory Rel. Fields (2009), DOI 10.1007/s00440-009-0250-6.
- E. Pardoux and A. Yu. Veretennikov, On Poisson equation and diffusion approximation 2, Ann. Prob. 31 (2003), 1166–1192. MR 1988467 (2004d:60156)
- A. M. Kulik and A. Yu. Veretennikov, On extended Poisson equation and diffusion approximation (2011). (to appear) MR 2933700 (2012m:60128)
- H. Kunita, Absolute continuity of Markov processes and generators, Nagoya Math. J. 36 (1969), 1–26. MR 0250387 (40:3626)
- D. Revuz and M. and Yor, Continuous Martingales and Brownian Motion, third edition, Springer-Verlag, Berlin, 1999. MR 1725357 (2000h:60050)
- Z. Palmowski and T. Rolski, A technique for exponential change of measure for Markov processes, Bernoulli 8 (2002), no. 6, 767–785. MR 1963661 (2004f:60152)
- V. M. Zolotarev, Modern theory of summation of random variables, “Nauka”, Moscow, 1986; English transl., VSP, Utrecht, 1997. MR 1640024 (99m:60002)
- R. M. Dudley, Real Analysis and Probability, Cambridge University Press, Cambridge, 2002. MR 1932358 (2003h:60001)
- N. Ikeda and S. Watanabe, Stochastic differential equations and diffusion processes, North-Holland Publishing Co. and Kodansha, Ltd., Amsterdam–New York–Tokyo, 1981. MR 637061 (84b:60080)
- R. L. Dobrushin, Prescribing a system of random variables by conditional distributions, Teor. Verojatnost. i Primenen. 15 (1970), no. 3, 469–497; English transl. in Theory Probab. Appl. 15 (1970), no. 3, 458–486. MR 0298716 (45:7765)
- I. I. Gihman and A. V. Skorohod, Stochastic Differential Equations, “Naukova Dumka”, Kiev, 1968; English transl., Springer-Verlag, Berlin–Heidelberg–New York, 1972. MR 0346904 (49:11625)
Similar Articles
Retrieve articles in Theory of Probability and Mathematical Statistics
with MSC (2010):
60H10,
60J10
Retrieve articles in all journals
with MSC (2010):
60H10,
60J10
Additional Information
A. Yu. Veretennikov
Affiliation:
School of Mathematics, University of Leeds, LS2 9JT, Leeds, United Kingdom; Institute of Information Transmission Problems, Bol’shoi Karetny Street 19, 127994, Moscow, Russia
Email:
A.Veretennikov@leeds.ac.uk
A. M. Kulik
Affiliation:
Institute of Mathematics, National Academy of Science of Ukraine, Tereshchenkivs’ka Street, 3, 01601 Kyiv–4, Ukraine
Email:
kulik@imath.kiev.ua
Keywords:
Markov process,
extended generator,
Poisson equation,
Doob–Meyer decomposition
Received by editor(s):
September 6, 2011
Published electronically:
January 11, 2013
Additional Notes:
This research was partially supported by the State Fund for Fundamental Researches, project $\Phi$40.1/023
This paper is based on the talk presented at the International Conference “Modern Stochastics: Theory and Applications II” held on September 7–11, 2010, at Kyiv National Taras Shevchenko University and dedicated to the anniversaries of the prominent Ukrainian scientists Anatoliĭ Skorokhod, Vladimir Korolyuk, and Igor Kovalenko
Article copyright:
© Copyright 2013
American Mathematical Society