The Annals of Probability

Random perturbations of nonlinear oscillators

Mark Freidlin and Matthias Weber

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Abstract

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

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

Dates
First available in Project Euclid: 31 May 2002

Permanent link to this document
https://projecteuclid.org/euclid.aop/1022855739

Digital Object Identifier
doi:10.1214/aop/1022855739

Mathematical Reviews number (MathSciNet)
MR1634409

Zentralblatt MATH identifier
0935.60038

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

Keywords
Averaging principle random perturbations Hamiltonian systems

Citation

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


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