# American Institute of Mathematical Sciences

March  2013, 18(2): 523-549. doi: 10.3934/dcdsb.2013.18.523

## On large deviations in the averaging principle for SDE's with a "full dependence,'' revisited

 1 School of Mathematics, University of Leeds, LS2 9JT, United Kingdom

Received  November 2011 Revised  May 2012 Published  November 2012

We establish a large deviation principle for stochastic differential equations with averaging in the case when all coefficients of the fast component depend on the slow one, including diffusion.
Citation: Alexander Veretennikov. On large deviations in the averaging principle for SDE's with a "full dependence,'' revisited. Discrete & Continuous Dynamical Systems - B, 2013, 18 (2) : 523-549. doi: 10.3934/dcdsb.2013.18.523
##### References:
 [1] M. I. Freidlin, Fluctuations in dynamical systems with averaging, Dok. Acad. Nauk SSSR, 226 (1976), 273-276 (in Russian).  Google Scholar [2] M. I. Freidlin, Averaging principle and large deviations, Uspekhi Matem. Nauk, 33 (1978) 107-160. (in Russian).  Google Scholar [3] M. I. Freidlin and A. D. Wentzell, "Random Perturbations of Dynamical Systems,'' Springer-Verlag, New York, 1984.  Google Scholar [4] O. V. Gulinsky and A. Yu. Veretennikov, "Large Deviations for Discrete-Time Processes with Averaging,'' VSP, Utrecht, 1993.  Google Scholar [5] N. Ikeda and S. Watanabe, "Stochastic Differential Equations and Diffusion Processes,'' $2^{nd}$ edition, North-Holland, Amsterdam, 1989.  Google Scholar [6] T. Kato, "Perturbation Theory for Linear Operators,'' $2^{nd}$ edition, Springer-Verlag, New York, 1976.  Google Scholar [7] Yu. Kifer, "Large Deviations and Adiabatic Transitions for Dynamical Systems and Markov Processes in Fully Coupled Averaging,'' Memoirs of the Amer. Math. Soc. 944, AMS, Providence, RI, 2009.  Google Scholar [8] M. A. Krasnosel'skii, E. A. Lifshitz and A. V. Sobolev, "Positive Linear Systems,'' Helderman, Berlin, 1989.  Google Scholar [9] N. V. Krylov, "Introduction to the Theory of Random Processes,'' AMS, Providence, RI, 1995.  Google Scholar [10] R. S. Liptser, Large deviations for two scaled diffusions, Probability Theory and Related Fields, 106(1) (1996), 71-104; preprint version (2005) at arXiv:math/0510029.  Google Scholar [11] R. Liptser, V. Spokoiny and A. Yu. Veretennikov, Freidlin-Wentzell type large deviations for smooth processes, Markov Processes and Related Fields, 8 (2002), 611-636.  Google Scholar [12] R. T. Rockafellar, "Convex Analysis,'' Princeton Univ. Press., Princeton, NJ, 1970.  Google Scholar [13] A. Yu. Veretennikov, On large deviations in the averaging principle for stochastic differential equations with periodic coefficients 2, Math. USSR Izvestiya, 39 (1992), 677-701.  Google Scholar [14] A. Yu. Veretennikov, Large deviations in averaging principle for stochastic differential equation systems (noncompact case), Stochastics Stochastics Rep., 48 (1994), 83-96.  Google Scholar [15] A. Yu. Veretennikov, On large deviations for stochastic differential equations with a small diffusion and averaging, Theory Probab. Appl., 43 (1998), 335-337. Google Scholar [16] A. Yu. Veretennikov, On large deviations in the averaging principle for SDE's with a "full dependence'', Ann. Probab., 27(1) (1999) 284-296.  Google Scholar [17] A. Yu. Veretennikov, On large deviations in the averaging principle for SDE's with a "full dependence'', correction,, preprint, ().  ,+" target="_new" title="Go to article in Google Scholar"> Google Scholar

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##### References:
 [1] M. I. Freidlin, Fluctuations in dynamical systems with averaging, Dok. Acad. Nauk SSSR, 226 (1976), 273-276 (in Russian).  Google Scholar [2] M. I. Freidlin, Averaging principle and large deviations, Uspekhi Matem. Nauk, 33 (1978) 107-160. (in Russian).  Google Scholar [3] M. I. Freidlin and A. D. Wentzell, "Random Perturbations of Dynamical Systems,'' Springer-Verlag, New York, 1984.  Google Scholar [4] O. V. Gulinsky and A. Yu. Veretennikov, "Large Deviations for Discrete-Time Processes with Averaging,'' VSP, Utrecht, 1993.  Google Scholar [5] N. Ikeda and S. Watanabe, "Stochastic Differential Equations and Diffusion Processes,'' $2^{nd}$ edition, North-Holland, Amsterdam, 1989.  Google Scholar [6] T. Kato, "Perturbation Theory for Linear Operators,'' $2^{nd}$ edition, Springer-Verlag, New York, 1976.  Google Scholar [7] Yu. Kifer, "Large Deviations and Adiabatic Transitions for Dynamical Systems and Markov Processes in Fully Coupled Averaging,'' Memoirs of the Amer. Math. Soc. 944, AMS, Providence, RI, 2009.  Google Scholar [8] M. A. Krasnosel'skii, E. A. Lifshitz and A. V. Sobolev, "Positive Linear Systems,'' Helderman, Berlin, 1989.  Google Scholar [9] N. V. Krylov, "Introduction to the Theory of Random Processes,'' AMS, Providence, RI, 1995.  Google Scholar [10] R. S. Liptser, Large deviations for two scaled diffusions, Probability Theory and Related Fields, 106(1) (1996), 71-104; preprint version (2005) at arXiv:math/0510029.  Google Scholar [11] R. Liptser, V. Spokoiny and A. Yu. Veretennikov, Freidlin-Wentzell type large deviations for smooth processes, Markov Processes and Related Fields, 8 (2002), 611-636.  Google Scholar [12] R. T. Rockafellar, "Convex Analysis,'' Princeton Univ. Press., Princeton, NJ, 1970.  Google Scholar [13] A. Yu. Veretennikov, On large deviations in the averaging principle for stochastic differential equations with periodic coefficients 2, Math. USSR Izvestiya, 39 (1992), 677-701.  Google Scholar [14] A. Yu. Veretennikov, Large deviations in averaging principle for stochastic differential equation systems (noncompact case), Stochastics Stochastics Rep., 48 (1994), 83-96.  Google Scholar [15] A. Yu. Veretennikov, On large deviations for stochastic differential equations with a small diffusion and averaging, Theory Probab. Appl., 43 (1998), 335-337. Google Scholar [16] A. Yu. Veretennikov, On large deviations in the averaging principle for SDE's with a "full dependence'', Ann. Probab., 27(1) (1999) 284-296.  Google Scholar [17] A. Yu. Veretennikov, On large deviations in the averaging principle for SDE's with a "full dependence'', correction,, preprint, ().  ,+" target="_new" title="Go to article in Google Scholar"> Google Scholar
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