The Smoluchowski–Kramers approximation for the stochastic Liénard equation by mean-field
Keyword(s):
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.
1991 ◽
Vol 23
(02)
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pp. 303-316
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1997 ◽
Vol 52
(2)
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pp. 327-340
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2007 ◽
Vol 21
(02n03)
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pp. 139-154
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2013 ◽
Vol 469
(2156)
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pp. 20130201
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1987 ◽
Vol 24
(02)
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pp. 370-377
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