Some Remarks on the Smoluchowski–Kramers Approximation

AbstractAccording to the Smoluchowski–Kramers approximation, solution qtμ of the equation $$\mu \ddot q_t^\mu = b(q_t^\mu ) - \dot q_t^\mu + \sigma (q_t^\mu )\dot W_t ,q_0 = q,\dot q = p$$ , where $$\dot W_t $$ is the White noise, converges to the solution of equation $$\dot q_t = b(q_t ) + \sigma (q_t )\dot W_t ,q_0 = q$$ as µ ↓ 0. Many asymptotic problems for the last equation were studied in recent years. We consider relations between asymptotics for the first order equation and the original second order equation. Homogenization, large deviations and stochastic resonance, approximation of Brownian motion Wt by a smooth stochastic process, stationary distributions are considered.