The Annals of Probability

On Poisson equation and diffusion approximation 2

È. Pardoux and A. Yu. Veretennikov

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Abstract

Three different results are established which turn out to be closely connected so that the first one implies the second one which in turn implies the third one. The first one states the smoothness of an invariant diffusion density with respect to a parameter. The second establishes a similar smoothness of the solution of the Poisson equation in $\mathbb{R}^d$. The third one states a diffusion approximation result, or in other words an averaging of singularly perturbed diffusion for "fully coupled SDE systems'' or "SDE systems with complete dependence.''

Article information

Source
Ann. Probab. Volume 31, Number 3 (2003), 1166-1192.

Dates
First available: 12 June 2003

Permanent link to this document
http://projecteuclid.org/euclid.aop/1055425774

Digital Object Identifier
doi:10.1214/aop/1055425774

Mathematical Reviews number (MathSciNet)
MR1988467

Subjects
Primary: 60H10: Stochastic ordinary differential equations [See also 34F05] 60J17 35J15: Second-order elliptic equations 35B30: Dependence of solutions on initial and boundary data, parameters [See also 37Cxx]

Keywords
Diffusion approximation Poisson equation with parameter invariant density with parameter Green function with parameter.

Citation

Pardoux, È.; Veretennikov, A. Yu. On Poisson equation and diffusion approximation 2. The Annals of Probability 31 (2003), no. 3, 1166--1192. doi:10.1214/aop/1055425774. http://projecteuclid.org/euclid.aop/1055425774.


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References

  • [1] BAKHVALOV, N. S. (1974). Averaged characteristics of bodies with periodic structure. Soviet Phy s. Dokl. 19 650-651. [Translated from Dokl. Akad. Nauk SSSR 218 (1974) 1046-1048.]
  • [2] BAKHVALOV, N. S. (1975). Averaging of partial differential equations with rapidly oscillating coefficients. Sov. Math. Dokl. 16 351-355. [Translated from Dokl. Akad. Nauk SSSR 221 (1975) 516-519.]
  • [3] BENVENISTE, A., METIVIER, M. and PRIOURET, P. (1987). Algorithmes adaptifs et approximations stochastiques. Théorie et applications à l'identification, au traitment du signal et à reconnaissance des formes. Masson, Paris. [English translation (1990) Adaptive Algorithms and Stochastic Approximations. Springer, Berlin.]
  • [4] EIDELMAN, S. D. (1956). On the fundamental solutions of parabolic sy stems. Soviet Math. Sbornik 38 51-92. [English translation Amer. Math. Soc. Transl. 41 (1964) 1-48.]
  • [5] EIDELMAN, S. D. (1964). Parabolic Sy stems. Nauka, Moscow (in Russian).
  • [6] ETHIER, S. N. and KURTZ, T. G. (1986). Markov Processes. Characterization and Convergence. Wiley, New York.
  • [7] FREIDLIN, M. I. and WENTZELL, S. D. (1984). Random Perturbations of Dy namical Sy stems. Springer, New York.
  • [8] FRIEDMAN, A. (1964) Partial Differential Equations of Parabolic Ty pe. Prentice-Hall, Englewood Cliffs, NJ.
  • [9] GLy NN, P. W. and MEy N, S. P. (1996). A Liapounov bound for solutions of the Poisson equation. Ann. Probab. 24 916-931.
  • [10] KIFER, YU. (2001). Averaging and climate models. In Stochastic Climate Models (P. Imkeller and J. von Stroch, eds.). Birkhäuser, Basel.
  • [11] KRy LOV, N. V. (1980). Controlled Diffusion Processes. Springer, Berlin.
  • [12] LADy ZENSKAJA, O., SOLONNIKOV, V. and URAL'CEVA, N. (1968). Linear and Quasilinear Equations of Parabolic Ty pe. Amer. Math. Soc., Providence, RI.
  • [13] PAPANICOLAOU, G. C., STROOCK, D. W. and VARADHAN, S. R. S (1976). Martingale approach to some limit theorems. In Statistical Mechanics and Dy namical Sy stems (D. Ruelle ed.) 1-12. Duke Univ. Math. Series 3.
  • [14] PARDOUX, É. and VERETENNIKOV, A. YU. (2000). On regularity of an invariant density of a Markov chain in a parameter. Russian Math. Dokl. 370 158-160.
  • [15] PARDOUX, É. and VERETENNIKOV, A. YU. (2001). On Poisson equation and diffusion approximation 1. Ann. Probab. 29 1061-1085.
  • [16] VERETENNIKOV, A. YU. (1997). On poly nomial mixing bounds for stochastic differential equations. Stoch. Processes Appl. 70 115-127.
  • [17] VERETENNIKOV, A. YU. (1999). On poly nomial mixing and convergence rate for stochastic difference and differential equations. Teor. Veroy atnost. i Primenen. 44 312-327 (in Russian). [English translation Theory Probab. Appl. 44 (2000) 361-374.]
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