Asymptotic behaviour for a time-inhomogeneous Kolmogorov type diffusion

We study a kinetic stochastic model with a non-linear time-inhomogeneous drag force and a Brownian-type random force. More precisely, the Kolmogorov type diffusion [[EQUATION]] is considered : here [[EQUATION]] is the position of the particle and [[EQUATION]] is its velocity and is solution of a stochastic differential equation driven by a one-dimensional Brownian motion, with the drift of the form [[EQUATION]] . The function F satisfies some homogeneity condition and  [[EQUATION]] is positive. The behaviour of the process  in large time is proved by using stochastic analysis tools.