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

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  • [1] V. I. Arnold, Mathematical methods of classical mechanics, Springer-Verlag, 1970. MR 997295 (90c:58046)
  • [2] M. Berger, Private communication.
  • [3] R. N. Bhattacharya, Criteria for recurrence and existence of invariant measures for multidimensional diffusions, Ann. Probab. 6 (1978), 541-553. MR 0494525 (58:13375)
  • [4] A. Friedman, Stochastic differential equations and application, Vol. I, Academic Press, 1976.
  • [5] E. Hille, and R. Phillips, Functional analysis and semigroups, rev. ed., Amer. Math. Soc., Providence, R.I., 1957. MR 0089373 (19:664d)
  • [6] R. Z. Khas'minskii, Ergodic properties of recurrent diffusion process and stabilization of the solution of the Cauchy problem for parabolic equations, Theory Probab. Appl. 5 (1960), 179-191. MR 0133871 (24:A3695)
  • [7] -, Principle of averaging for parabolic and elliptic differential equations and for Markov process with small diffusion, Theory. Probab. Appl. 7 (1963), 1-21.
  • [8] -, The behavior of a self oscillating system acted upon by slight noise, Prikl. Mat. Mek. 27 (1963), 683-687. MR 0162032 (28:5234)
  • [9] -, The behavior of a conservative system under the action of a slight friction and slight random noise, 28 (1964), 931-935. MR 0191711 (32:9113)
  • [10] T. G. Kurtz, A limit theorem for perturbed operator semigroups with applications to random evolution, J. Funct. Anal. 12 (1973), 55-67. MR 0365224 (51:1477)
  • [11] R. Pinsky, Private communication.
  • [12] G. Wolansky, Stochastic perturbations to conservative dynamical systems on the plane. II: Recurrency conditions, Trans. Amer. Math. Soc. 309 (1988), 641-657. MR 961605 (90b:35253)
  • [13] -, Elliptic perturbations of nonlinear oscillations in the presence of resonances, Indiana J. Math. (to appear).

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Keywords: Diffusion process, perturbation, invariant distribution
Article copyright: © Copyright 1988 American Mathematical Society

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