  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 $(V,X)$ is considered: here $X$ is the position of the particle and
$V$ is its velocity and is solution of a stochastic differential equation
driven by a one-dimensional Brownian motion, with the drift of the form
$t^{-\beta}F(v)$. The function $F$ satisfies some homogeneity condition and
$\beta$ is positive. The behaviour of the process $(V,X)$ in large time is
proved by using stochastic analysis tools.
