Stochastic perturbations to conservative dynamical systems on the plane. I. Convergence of invariant distributions
HTML articles powered by AMS MathViewer
- by G. Wolansky PDF
- 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.References
- V. I. Arnol′d, Mathematical methods of classical mechanics, 2nd ed., Graduate Texts in Mathematics, vol. 60, Springer-Verlag, New York, 1989. Translated from the Russian by K. Vogtmann and A. Weinstein. MR 997295, DOI 10.1007/978-1-4757-2063-1 M. Berger, Private communication.
- R. N. Bhattacharya, Criteria for recurrence and existence of invariant measures for multidimensional diffusions, Ann. Probab. 6 (1978), no. 4, 541–553. MR 0494525, DOI 10.1214/aop/1176995476 A. Friedman, Stochastic differential equations and application, Vol. I, Academic Press, 1976.
- Einar Hille and Ralph S. Phillips, Functional analysis and semi-groups, American Mathematical Society Colloquium Publications, Vol. 31, American Mathematical Society, Providence, R.I., 1957. rev. ed. MR 0089373
- R. Z. Has′minskiĭ, Ergodic properties of recurrent diffusion processes and stabilization of the solution of the Cauchy problem for parabolic equations, Teor. Verojatnost. i Primenen. 5 (1960), 196–214 (Russian, with English summary). MR 0133871 —, Principle of averaging for parabolic and elliptic differential equations and for Markov process with small diffusion, Theory. Probab. Appl. 7 (1963), 1-21.
- R. Z. Khas’minskii, The behavior of a self-oscillating system acted upon by slight noise, J. Appl. Math. Mech. 27 (1963), 1035–1044. MR 0162032, DOI 10.1016/0021-8928(63)90184-9
- R. Z. Khas’minskii, The behavior of a conservative system under the action of slight friction and slight random noise, J. Appl. Math. Mech. 28 (1964), 1126–1130 (1965). MR 0191711, DOI 10.1016/0021-8928(64)90017-6
- Thomas G. Kurtz, A limit theorem for perturbed operator semigroups with applications to random evolutions, J. Functional Analysis 12 (1973), 55–67. MR 0365224, DOI 10.1016/0022-1236(73)90089-x R. Pinsky, Private communication.
- G. Wolansky, Stochastic perturbations to conservative dynamical systems on the plane. II. Recurrency conditions, Trans. Amer. Math. Soc. 309 (1988), no. 2, 641–657. MR 961605, DOI 10.1090/S0002-9947-1988-0961605-1 —, Elliptic perturbations of nonlinear oscillations in the presence of resonances, Indiana J. Math. (to appear).
Additional Information
- © Copyright 1988 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 309 (1988), 621-639
- MSC: Primary 35R60; Secondary 58F11, 60J60, 93E03
- DOI: https://doi.org/10.1090/S0002-9947-1988-0961604-X
- MathSciNet review: 961604