Anomalous diffusion behaviour for a time-inhomogeneous Kolmogorov type diffusion

Mihai Gradinaru (1) , Emeline Luirard (1)

1 IRMAR - Institut de Recherche Mathématique de Rennes

We consider a kinetic stochastic model with a non-linear time-inhomogeneous drag force and a Brownian random force. More precisely, we study the couple position $X_t$ of a particle and its velocity which is a solution of a stochastic differential equation driven by a one-dimensional Brownian motion, with the drift of the form $t^{-\beta}F(v)$, $F$ satisfying some homogeneity condition and $\beta>0$. The behaviour of $(V,X)$ in large time is proven and the precise rate of convergence is pointed out by using stochastic analysis tools.

Mots clés: scaling transformations; explosion times; kinetic stochastic equation; time-inhomogeneous diffusions; asymptotic distributions

Domaines: Probabilités [math.PR]

Mihai Gradinaru, Emeline Luirard. Anomalous diffusion behaviour for a time-inhomogeneous Kolmogorov type diffusion. 2020. ⟨hal-02552764v2⟩