The Annals of Probability

Random perturbations of nonlinear oscillators

Mark Freidlin and Matthias Weber

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Degenerate white noise perturbations of Hamiltonian systems in $R^2$ are studied. In particular, perturbations of a nonlinear oscillator with 1 degree of freedom are considered. If the oscillator has more than one stable equilibrium, the long time behavior of the system is defined by a diffusion process on a graph. Inside the edges the process is defined by a standard averaging procedure, but to define the process for all $t > 0$ one should add gluing conditions at the vertices. Calculation of the gluing conditions is based on delicate Hörmander-type estimates.

Article information

Ann. Probab., Volume 26, Number 3 (1998), 925-967.

First available in Project Euclid: 31 May 2002

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60H10: Stochastic ordinary differential equations [See also 34F05] 34C29: Averaging method 35B20: Perturbations
Secondary: 35H05

Averaging principle random perturbations Hamiltonian systems


Freidlin, Mark; Weber, Matthias. Random perturbations of nonlinear oscillators. Ann. Probab. 26 (1998), no. 3, 925--967. doi:10.1214/aop/1022855739.

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  • [1] Adams, R. A. (1975). Sobolev Spaces. Academic Press, New York.
  • [2] Baskakova, P. E. and Borodin, A. N. (1995). Tables of distributions of functionals of Brownian motion. J. Math. Sci. 75 1873-1883.
  • [3] Burdzeiko, B., Ignat'ev, Yu., Khasminskii, R. and B. Shahgildiam (1979). Statistical analysis of a model of phase synchronization. In Proceedings of the International Conference on Information Theory, Tbilisi, 64-67 (in Russian).
  • [4] Feller, W. (1954). Diffusion processes in one dimension. Trans. Amer. Math. Soc. 97 1-31.
  • [5] Freidlin, M. (1985). Functional Integration and Partial Differential Equations. Princeton Univ. Press.
  • [6] Freidlin, M. I. and Wentzell, A. D. (1993). Diffusion processes on graphs and the averaging principle. Ann. Probab. 21 2215-2245.
  • [7] Freidlin, M. I. and Wentzell, A. D. (1994). Random perturbations of Hamiltonian systems. Mem. Amer. Math. Soc. 109 (523).
  • [8] H ¨ormander, L. (1985). The Analysis of Linear Partial Differential Operators 3. Springer, Berlin.
  • [9] Oleinik, A. O. and Radkevic, E. V. (1973). Second Order Equations with Nonnegative Characteristic Form. Plenum, New York.