Asymptotic behavior for a time-inhomogeneous Kolmogorov type diffusion

Abstract

We study a kinetic stochastic model with a non-linear time-inhomogeneous friction force and a Brownian-type random force. More precisely, a Kolmogorov type diffusion (V, X) is considered: here, X is the position of the particle, and V is its velocity. The process V is solution to a stochastic differential equation driven by a one-dimensional Brownian motion, with a drift of the form t−βF(v). The function F satisfies some homogeneity condition, and β is a real number. The behavior in large time of the process (V, X) is proved by using stochastic analysis tools.