Electronic Communications in Probability

Double averaging principle for periodically forced stochastic slow-fast systems

Gilles Wainrib

Full-text: Open access


This paper is devoted to obtaining an averaging principle for systems of slow-fast stochastic differential equations, where the fast variable drift is periodically modulated on a fast time-scale. The approach developed here combines probabilistic methods with a recent analytical result on long-time behavior for second order elliptic equations with time-periodic coefficients.

Article information

Electron. Commun. Probab., Volume 18 (2013), paper no. 51, 12 pp.

Accepted: 26 June 2013
First available in Project Euclid: 7 June 2016

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 70K70: Systems with slow and fast motions
Secondary: 65C30: Stochastic differential and integral equations

averaging principle slow-fast stochastic differential equation periodic averaging inhomogeneous Markov process

This work is licensed under a Creative Commons Attribution 3.0 License.


Wainrib, Gilles. Double averaging principle for periodically forced stochastic slow-fast systems. Electron. Commun. Probab. 18 (2013), paper no. 51, 12 pp. doi:10.1214/ECP.v18-1975. https://projecteuclid.org/euclid.ecp/1465315590

Export citation


  • Cerrai, Sandra. A Khasminskii type averaging principle for stochastic reaction-diffusion equations. Ann. Appl. Probab. 19 (2009), no. 3, 899–948.
  • M. Galtier and G. Wainrib, Multiscale analysis of slow-fast neuronal learning models with noise., The Journal of Mathematical Neurosciences, 2,13 (2012).
  • Givon, Dror. Strong convergence rate for two-time-scale jump-diffusion stochastic differential systems. Multiscale Model. Simul. 6 (2007), no. 2, 577–594 (electronic).
  • Karatzas, Ioannis; Shreve, Steven E. Brownian motion and stochastic calculus. Second edition. Graduate Texts in Mathematics, 113. Springer-Verlag, New York, 1991. xxiv+470 pp. ISBN: 0-387-97655-8
  • R.Z. Khas' minskii, The averaging principle for stochastic differential equations, Problemy Peredachi Informatsii 4 (1968), no. 2, 86–87.
  • Kifer, Yuri. Large deviations and adiabatic transitions for dynamical systems and Markov processes in fully coupled averaging. Mem. Amer. Math. Soc. 201 (2009), no. 944, viii+129 pp. ISBN: 978-0-8218-4425-0
  • Liu, Di. Strong convergence of principle of averaging for multiscale stochastic dynamical systems. Commun. Math. Sci. 8 (2010), no. 4, 999–1020.
  • Lorenzi, Luca; Lunardi, Alessandra; Zamboni, Alessandro. Asymptotic behavior in time periodic parabolic problems with unbounded coefficients. J. Differential Equations 249 (2010), no. 12, 3377–3418.
  • Papanicolaou, George C. Some probabilistic problems and methods in singular perturbations. Summer Research Conference on Singular Perturbations: Theory and Applications (Northern Arizona Univ., Flagstaff, Ariz., 1975). Rocky Mountain J. Math. 6 (1976), no. 4, 653–674.
  • Veretennikov, A. Yu. On large deviations in the averaging principle for stochastic differential equations with periodic coefficients. II. (Russian) Izv. Akad. Nauk SSSR Ser. Mat. 55 (1991), no. 4, 691–715; translation in Math. USSR-Izv. 39 (1992), no. 1, 677–701
  • Veretennikov, A. Yu. On an averaging principle for systems of stochastic differential equations. (Russian) Mat. Sb. 181 (1990), no. 2, 256–268; translation in Math. USSR-Sb. 69 (1991), no. 1, 271–284
  • G. Wainrib, Noise-controlled dynamics through the averaging principle for stochastic slow-fast systems., Phys. Rev. E (2011).
  • G. Wainrib; M. Thieullen; K. Pakdaman, Reduction of stochastic conductance-based neuron models with time-scales separation. J. Comput. Neurosci. 32 (2012), no. 2, 327–346.
  • E, Weinan; Liu, Di; Vanden-Eijnden, Eric. Analysis of multiscale methods for stochastic differential equations. Comm. Pure Appl. Math. 58 (2005), no. 11, 1544–1585.