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

Wolansky, G.

doi:10.1090/S0002-9947-1988-0961604-X

Stochastic perturbations to conservative dynamical systems on the plane. I. Convergence of invariant distributions
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Trans. Amer. Math. Soc. 309 (1988), 621-639 Request permission

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 ${L_2}$ convergence of the associated densities is proved.

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