American Institute of Mathematical Sciences

Advanced
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

Alexander Veretennikov 1,

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.
Keywords: Large deviations, full dependence., averaging principle, stochastic differential equation.
Mathematics Subject Classification: Primary: 60F10; Secondary: 60H1.
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.   Google Scholar

[2]

M. I. Freidlin, Averaging principle and large deviations,, Uspekhi Matem. Nauk, 33 (1978), 107.   Google Scholar

[3]

M. I. Freidlin and A. D. Wentzell, "Random Perturbations of Dynamical Systems,'', Springer-Verlag, (1984).   Google Scholar

[4]

O. V. Gulinsky and A. Yu. Veretennikov, "Large Deviations for Discrete-Time Processes with Averaging,'', VSP, (1993).   Google Scholar

[5]

N. Ikeda and S. Watanabe, "Stochastic Differential Equations and Diffusion Processes,'', $2^{nd}$ edition, (1989).   Google Scholar

[6]

T. Kato, "Perturbation Theory for Linear Operators,'', $2^{nd}$ edition, (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, (2009).   Google Scholar

[8]

M. A. Krasnosel'skii, E. A. Lifshitz and A. V. Sobolev, "Positive Linear Systems,'', Helderman, (1989).   Google Scholar

[9]

N. V. Krylov, "Introduction to the Theory of Random Processes,'', AMS, (1995).   Google Scholar

[10]

R. S. Liptser, Large deviations for two scaled diffusions,, Probability Theory and Related Fields, 106(1) (1996), 71.   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.   Google Scholar

[12]

R. T. Rockafellar, "Convex Analysis,'', Princeton Univ. Press., (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.   Google Scholar

[14]

A. Yu. Veretennikov, Large deviations in averaging principle for stochastic differential equation systems (noncompact case),, Stochastics Stochastics Rep., 48 (1994), 83.   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.   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.   Google Scholar

[17]

A. Yu. Veretennikov, On large deviations in the averaging principle for SDE's with a "full dependence'', correction,, preprint, ().   Google Scholar

show all references

References:
[1]

M. I. Freidlin, Fluctuations in dynamical systems with averaging,, Dok. Acad. Nauk SSSR, 226 (1976), 273.   Google Scholar

[2]

M. I. Freidlin, Averaging principle and large deviations,, Uspekhi Matem. Nauk, 33 (1978), 107.   Google Scholar

[3]

M. I. Freidlin and A. D. Wentzell, "Random Perturbations of Dynamical Systems,'', Springer-Verlag, (1984).   Google Scholar

[4]

O. V. Gulinsky and A. Yu. Veretennikov, "Large Deviations for Discrete-Time Processes with Averaging,'', VSP, (1993).   Google Scholar

[5]

N. Ikeda and S. Watanabe, "Stochastic Differential Equations and Diffusion Processes,'', $2^{nd}$ edition, (1989).   Google Scholar

[6]

T. Kato, "Perturbation Theory for Linear Operators,'', $2^{nd}$ edition, (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, (2009).   Google Scholar

[8]

M. A. Krasnosel'skii, E. A. Lifshitz and A. V. Sobolev, "Positive Linear Systems,'', Helderman, (1989).   Google Scholar

[9]

N. V. Krylov, "Introduction to the Theory of Random Processes,'', AMS, (1995).   Google Scholar

[10]

R. S. Liptser, Large deviations for two scaled diffusions,, Probability Theory and Related Fields, 106(1) (1996), 71.   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.   Google Scholar

[12]

R. T. Rockafellar, "Convex Analysis,'', Princeton Univ. Press., (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.   Google Scholar

[14]

A. Yu. Veretennikov, Large deviations in averaging principle for stochastic differential equation systems (noncompact case),, Stochastics Stochastics Rep., 48 (1994), 83.   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.   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.   Google Scholar

[17]

A. Yu. Veretennikov, On large deviations in the averaging principle for SDE's with a "full dependence'', correction,, preprint, ().   Google Scholar

[1]

Yongqiang Suo, Chenggui Yuan. Large deviations for neutral stochastic functional differential equations. Communications on Pure & Applied Analysis, 2020, 19 (4) : 2369-2384. doi: 10.3934/cpaa.2020103
[2]

Yueling Li, Yingchao Xie, Xicheng Zhang. Large deviation principle for stochastic heat equation with memory. Discrete & Continuous Dynamical Systems - A, 2015, 35 (11) : 5221-5237. doi: 10.3934/dcds.2015.35.5221
[3]

Peng Gao. Averaging principle for stochastic Kuramoto-Sivashinsky equation with a fast oscillation. Discrete & Continuous Dynamical Systems - A, 2018, 38 (11) : 5649-5684. doi: 10.3934/dcds.2018247
[4]

Salah-Eldin A. Mohammed, Tusheng Zhang. Large deviations for stochastic systems with memory. Discrete & Continuous Dynamical Systems - B, 2006, 6 (4) : 881-893. doi: 10.3934/dcdsb.2006.6.881
[5]

Wenqing Hu, Chris Junchi Li. A convergence analysis of the perturbed compositional gradient flow: Averaging principle and normal deviations. Discrete & Continuous Dynamical Systems - A, 2018, 38 (10) : 4951-4977. doi: 10.3934/dcds.2018216
[6]

Bin Pei, Yong Xu, Yuzhen Bai. Convergence of p-th mean in an averaging principle for stochastic partial differential equations driven by fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1141-1158. doi: 10.3934/dcdsb.2019213
[7]

Xiaobin Sun, Jianliang Zhai. Averaging principle for stochastic real Ginzburg-Landau equation driven by $ \alpha $-stable process. Communications on Pure & Applied Analysis, 2020, 19 (3) : 1291-1319. doi: 10.3934/cpaa.2020063
[8]

Martino Bardi, Annalisa Cesaroni, Daria Ghilli. Large deviations for some fast stochastic volatility models by viscosity methods. Discrete & Continuous Dynamical Systems - A, 2015, 35 (9) : 3965-3988. doi: 10.3934/dcds.2015.35.3965
[9]

Yong Xu, Rong Guo, Di Liu, Huiqing Zhang, Jinqiao Duan. Stochastic averaging principle for dynamical systems with fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2014, 19 (4) : 1197-1212. doi: 10.3934/dcdsb.2014.19.1197
[10]

Anna Amirdjanova, Jie Xiong. Large deviation principle for a stochastic navier-Stokes equation in its vorticity form for a two-dimensional incompressible flow. Discrete & Continuous Dynamical Systems - B, 2006, 6 (4) : 651-666. doi: 10.3934/dcdsb.2006.6.651
[11]

Federico Bassetti, Lucia Ladelli. Large deviations for the solution of a Kac-type kinetic equation. Kinetic & Related Models, 2013, 6 (2) : 245-268. doi: 10.3934/krm.2013.6.245
[12]

Miguel Abadi, Sandro Vaienti. Large deviations for short recurrence. Discrete & Continuous Dynamical Systems - A, 2008, 21 (3) : 729-747. doi: 10.3934/dcds.2008.21.729
[13]

Markus Riedle, Jianliang Zhai. Large deviations for stochastic heat equations with memory driven by Lévy-type noise. Discrete & Continuous Dynamical Systems - A, 2018, 38 (4) : 1983-2005. doi: 10.3934/dcds.2018080
[14]

Boling Guo, Yan Lv, Wei Wang. Schrödinger limit of weakly dissipative stochastic Klein--Gordon--Schrödinger equations and large deviations. Discrete & Continuous Dynamical Systems - A, 2014, 34 (7) : 2795-2818. doi: 10.3934/dcds.2014.34.2795
[15]

Shihu Li, Wei Liu, Yingchao Xie. Large deviations for stochastic 3D Leray-$ \alpha $ model with fractional dissipation. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2491-2509. doi: 10.3934/cpaa.2019113
[16]

Dongmei Zheng, Ercai Chen, Jiahong Yang. On large deviations for amenable group actions. Discrete & Continuous Dynamical Systems - A, 2016, 36 (12) : 7191-7206. doi: 10.3934/dcds.2016113
[17]

Peng Gao, Yong Li. Averaging principle for the Schrödinger equations†. Discrete & Continuous Dynamical Systems - B, 2017, 22 (6) : 2147-2168. doi: 10.3934/dcdsb.2017089
[18]

Jesus Ildefonso Díaz, Jacqueline Fleckinger-Pellé. Positivity for large time of solutions of the heat equation: the parabolic antimaximum principle. Discrete & Continuous Dynamical Systems - A, 2004, 10 (1&2) : 193-200. doi: 10.3934/dcds.2004.10.193
[19]

Thomas Bogenschütz, Achim Doebler. Large deviations in expanding random dynamical systems. Discrete & Continuous Dynamical Systems - A, 1999, 5 (4) : 805-812. doi: 10.3934/dcds.1999.5.805
[20]

S. L. Ma'u, P. Ramankutty. An averaging method for the Helmholtz equation. Conference Publications, 2003, 2003 (Special) : 604-609. doi: 10.3934/proc.2003.2003.604

2018 Impact Factor: 1.008

Tools

Metrics

  • PDF downloads (21)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences