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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.

References

Chandrasekhar, S. (1954) Stochastic problems in physics and astronomy. In Selected Papers on Noise and Stochastic Processes , ed. Wax, N., Dover, New York.Google Scholar
Kramers, H. A. (1940) Brownian motion in a field of force and the diffusion model of chemical reactions. Physica 7, 284–304.CrossRefGoogle Scholar
Leonard, C. (1986) Une loi des grands nombres pour des systèmes de diffusions avec interaction et à coefficients non bornés. Ann. Inst. H. Poincaré, Prob. Statist. 22, 237–262.Google Scholar
Mckean, H. P. Jr. (1967) Propagation of chaos for a class of non-linear parabolic equations. Stochastic Differential Equations , Lecture Series in Differential Equations 7, Catholic University, 41–57.Google Scholar
Narita, K. (1989) Stochastic Liénard equation with mean-field interaction. SIAM J. Appl. Math. 49, 888–905.Google Scholar
Papanicolaou, G. C. (1975) Introduction to Asymptotic Analysis of Stochastic Equations. Lecture Notes, AMS Seminar, Renselaer Polytechnic Institute, Troy, NY.Google Scholar
Schuss, Z. (1980) Theory and Applications of Stochastic Differential Equations. Wiley, New York.Google Scholar
Smoluchowski, M. (1916) Drei Vortrage über Diffusion, Brownsche Bewegung und Koagulation von Kolloidteilchen. Phys. Z. 17, 557–585.Google Scholar