Stochastic perturbations to conservative dynamical systems on the plane. I. Convergence of invariant distributions

Author:
G. Wolansky

Journal:
Trans. Amer. Math. Soc. **309** (1988), 621-639

MSC:
Primary 35R60; Secondary 58F11, 60J60, 93E03

MathSciNet review:
961604

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Abstract: We consider a nonlinear system on the plane, given by an oscillator with homoclinic orbits. The above system is subjected to a perturbation, composed of a deterministic part and a random (white noise) part. Assuming the existence of a finite, invariant measure to the perturbed system, we deal with the convergence of the measures to a limit measure, as the perturbation parameter tends to zero. The limit measure is constructed in terms of the action function of the unperturbed oscillator, and the strong local convergence of the associated densities is proved.

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1988-0961604-X

Keywords:
Diffusion process,
perturbation,
invariant distribution

Article copyright:
© Copyright 1988
American Mathematical Society