Diffusion approximation of systems with weakly ergodic Markov perturbations. I
Authors:
A. Yu. Veretennikov and A. M. Kulik
Translated by:
N. Semenov
Journal:
Theor. Probability and Math. Statist. 87 (2013), 13-29
MSC (2010):
Primary 60H10, 60J10
DOI:
https://doi.org/10.1090/S0094-9000-2014-00901-1
Published electronically:
March 21, 2014
MathSciNet review:
3241443
Full-text PDF Free Access
Abstract |
References |
Similar Articles |
Additional Information
Abstract: Diffusion approximation type results are obtained for a system perturbed by a Markov process whose transition probabilities converge to the invariant distribution nonuniformly with respect to the initial value. In general, the mode of convergence is weaker than the total variation convergence.
- 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
- 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
- A. Yu. Veretennikov and A. M. Kulik, Diffusion approximation of systems with weakly ergodic Markov perturbations. II, Teor. Ĭmovīr. Mat. Stat. 88 (2013), 1–16 (Russian, with English, Russian and Ukrainian summaries); English transl., Theory Probab. Math. Statist. 88 (2014), 1–17. MR 3112631, DOI https://doi.org/10.1090/S0094-9000-2014-00915-1
- 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
- G. C. Papanicolaou, D. Stroock, and S. R. S. Varadhan, Martingale approach to some limit theorems, Papers from the Duke Turbulence Conference (Duke Univ., Durham, N.C., 1976), Paper No. 6, Duke Univ., Durham, N.C., 1977, pp. ii+120 pp. Duke Univ. Math. Ser., Vol. III. MR 0461684
- I. I. Gikhman and A. V. Skorokhod, Stokhasticheskie differentsial′nye uravneniya i ikh prilozheniya, “Naukova Dumka”, Kiev, 1982 (Russian). MR 678374
- 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, Teor. Veroyatnost. i Primenen. 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. Koroliuk 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 the Poisson equation and diffusion approximation. I, Ann. Probab. 29 (2001), 1061–1085. MR 1872736 (2002j:60120)
- E. Pardoux and A. Yu. Veretennikov, On Poisson equation and diffusion approximation. II, Ann. Probab. 31 (2003), 1166–1192. MR 1988467 (2004d:60156)
- A. Yu. Veretennikov and A. M. Kulik, Extended Poisson equation for weakly ergodic Markov processes, Teor. Imovir. Matem. Statyst. 85 (2011), 22–38; English transl. in Theor. Probability and Math. Statist. 85 (2012), 23–39. MR 2933700 (2012m:60128)
- A. Yu. Veretennikov and A. M. Kulik, Diffusion approximation of systems with weakly ergodic Markov perturbations. II, Teor. Ĭmovīr. Mat. Stat. No. 88 (2013), 1–16. MR 3112631
- 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 Univ. Press, 2002. MR 1932358 (2003h:60001)
- G. C. Papanicolaou, D. W. Stroock, and R. S. Varadhan, Martingale approach to some limit theorems (M. Reed, ed.), Conference on Statistical Mechanics, Dynamical Systems and Turbulence, Duke Univ. Press, 1977. MR 0461684 (57:1669)
- I. I. Gikhman and A. V. Skorokhod, Stochastic Differential Equations and their Applications, “Naukova dumka”, Kiev, 1982. (Russian) MR 678374 (84j:60003)
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, United Kingdom; Institute for Information Transmission Problems, Moscow, Russia
Email:
A.Veretennikov@leeds.ac.uk
A. M. Kulik
Affiliation:
Institute for Mathematics, National Academy of Science of Ukraine, Kyiv, Ukraine
Email:
kulik@imath.kiev.ua
Keywords:
Diffusion approximation,
corrector,
extended potential
Received by editor(s):
May 15, 2012
Published electronically:
March 21, 2014
Article copyright:
© Copyright 2014
American Mathematical Society