Diffusion approximation of systems with weakly ergodic Markov perturbations. II
Authors:
A. Yu. Veretennikov and A. M. Kulik
Translated by:
N. Semenov
Journal:
Theor. Probability and Math. Statist. 88 (2014), 1-17
MSC (2010):
Primary 60H10, 60J10
DOI:
https://doi.org/10.1090/S0094-9000-2014-00915-1
Published electronically:
July 24, 2014
MathSciNet review:
3112631
Full-text PDF Free Access
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Additional Information
Abstract: This paper is a continuation of the paper [A. Yu. Veretennikov and A. M. Kulik, Diffusion approximation for systems with weakly ergodic Markov perturbations. I, Theory Probab. Math. Statist. 87 (2012), 13–29]. Some corollaries of the general results are given in several particular cases being of their own interest. An example of a process being a solution of a stochastic differential equation with a Lévy noise is considered; we show that the assumptions imposed on the process can effectively be verified.
References
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References
- A. Yu. Veretennikov and A. M. Kulik, Diffusion approximation of systems with weakly ergodic Markov perturbations. I, Teor. Imovir. Mat. Stat. 87 (2012), 12–27; English transl. in Theory Probab. Math. Statist. 87 (2013), 13–29.
- T. Komorowski and A. Walczuk, Central limit theorem for Markov processes with spectral gap in the Wasserstein metric, Stochastic Process. Appl. 122 (2012), no. 5, 2155–2184. MR 2921976
- A. Yu. Veretennikov and A. M. Kulik, Extended Poisson equation for weakly ergodic Markov processes, Teor. Imovir. Mat. Stat. 85 (2011), 22–38; English transl. in Theory Probab. Math. Statist. 85, (2012) 23–39. MR 2933700 (2012m:60128)
- E. Pardoux and A. Yu. Veretennikov, On the Poisson equation and diffusion approximation. I, Ann. Probab. 29 (2001), 1061–1085. MR 1872736 (2002j:60120)
- R. L. Dobrushin, Prescribing a system of random variables by conditional distributions, Teor. Veroyatnost. Primenen. 15 (1970), no. 3, 469–497; English transl. in Theory Probab. Appl. 15 (1970), no. 3, 458–486. MR 0298716 (45:7765)
- A. Yu. Veretennikov, Coupling method for Markov chains under integral Doeblin type condition, Theory Stoch. Process. 8(24) (2002), no. 3–4, 383–391. MR 2027410 (2004j:60151)
- S. N. Ethier and T. G. Kurtz, Markov Processes. Characterization and Convergence, Wiley, New York, 1986. MR 838085 (88a:60130)
- V. S. Koroliuk and N. Limnios, Stochastic Systems in Merging Phase Space, World Scientific, New Jersey, 2005. MR 2205562 (2007a:60004)
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- A. Yu. Veretennikov, On polynomial mixing bounds for stochastic differential equations, Stoch. Process. Appl. 70 (1997), 115–127. MR 1472961 (99k:60158)
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- 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)
- N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, North Holland Publ. Co., Amsterdam–Oxford–New York, 1981. MR 637061 (84b:60080)
- A. M. Kulik and N. N. Leonenko, Ergodicity and mixing bounds for the Fisher–Snedecor diffusion, Bernoulli 19 (2013), no. 5B, 2153–2779. MR 3160555
- A. M. Kulik, Asymptotic and spectral properties of exponentially $\phi$-ergodic Markov processes, Stoch. Process. Appl. 121 (2011), 1044–1075. MR 2775106 (2012d:60194)
- M. Hairer and J. 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, Probab. Theory Related Fields 149 (2011), 223–259. MR 2773030
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Additional Information
A. Yu. Veretennikov
Affiliation:
School of Mathematics, University of Leeds, United Kingdom; Institute for Information Transmission Problems, Moscow, Russia
Email:
A.Veretennikov@leeds.ac.uk
A. M. Kulik
Affiliation:
Institute of Mathematics, National Academy of Science, Tereschenkivs’ka Street, 3, 01601, Kyiv, Ukraine
Email:
kulik@imath.kiev.ua
Keywords:
Diffusion approximation,
distance in variation,
Kantorovich–Rubinstein distance,
central limit theorem
Received by editor(s):
May 15, 2012
Published electronically:
July 24, 2014
Article copyright:
© Copyright 2014
American Mathematical Society