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Asymptotic behavior of velocity process in the Smoluchowski–Kramers approximation for stochastic differential equations

Published online by Cambridge University Press:  01 July 2016

Kiyomasa Narita
Kiyomasa Narita*
Affiliation:
Kanagawa University
*
∗Postal address: Department of Mathematics, Faculty of Technology, Kanagawa University, Rokkakubashi Kanagawa-ku, Yokohama 221, Japan.
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Abstract

Here a response of a non-linear oscillator of the Liénard type with a large parameter α ≥ 0 is formulated as a solution of a two-dimensional stochastic differential equation with mean-field of the McKean type. This solution is governed by a special form of the Fokker–Planck equation such as the Smoluchowski–Kramers equation, which is an equation of motion for distribution functions in position and velocity space describing the Brownian motion of particles in an external field. By a change of time and displacement we find that the velocity process converges to a one-dimensional Ornstein–Uhlenbeck process as α →∞.

Keywords

LIÉNARD OSCILLATORSMOLUCHOWSKI–KRAMERS EQUATIONMEAN-FIELDORNSTEIN–UHLENBECK PROCESS
Type
Research Article
Information
Advances in Applied Probability , Volume 23 , Issue 2 , June 1991 , pp. 317 - 326
Copyright
Copyright © Applied Probability Trust 1991 

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