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

The oscillator of the Liénard type with mean-field containing a large parameter . < 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 ., a one-dimensional limit diffusion process as . . .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.