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The Smoluchowski–Kramers approximation for the stochastic Liénard equation by mean-field

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

The oscillator of the Liénard type with mean-field containing a large parameter α < 0 is considered. The solution of the two-dimensional stochastic differential equation with mean-field of the McKean type is taken as the response of the oscillator. By a rigorous evaluation of the upper bound of the displacement process depending on the parameter α, a one-dimensional limit diffusion process as α → ∞is derived and identified. Then our result extends the Smoluchowski–Kramers approximation for the Langevin equation without mean-field to the McKean equation with mean-field.

Keywords

STOCHASTIC DIFFERENTIAL EQUATIONMCKEAN EQUATION
Type
Research Article
Information
Advances in Applied Probability , Volume 23 , Issue 2 , June 1991 , pp. 303 - 316
Copyright
Copyright © Applied Probability Trust 1991 

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Footnotes

Research supported in part by Grant-in-Aid for General Scientific Research (No. 1540203), Ministry of Education, Science and Culture.

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