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

Here a response of a non-linear oscillator of the Liénard type with a large parameter . . 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 . ...