Diffusion approximation of systems with weakly ergodic Markov perturbations. I

Veretennikov, A.; Kulik, A.

doi:10.1090/S0094-9000-2014-00901-1

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
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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.

References

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