Asymptotic behavior for a time-inhomogeneous stochastic differential equation driven by an α-stable Lévy process

Mihai Gradinaru (1) , Emeline Luirard (1)

1 IRMAR - Institut de Recherche Mathématique de Rennes

We study a one-dimensional kinetic stochastic model driven by a Lévy process, with a non-linear time-inhomogeneous drift. More precisely, the process $(V,X)$ is considered, where $X$ is the position of the particle and its velocity $V$ is the solution of a stochastic differential equation with a drift of the form $t^{-\beta}F(v)$. The driving noise can be a stable Lévy process of index $\alpha$ or a general Lévy process under appropriate assumptions. The function $F$ satisfies a homogeneity condition and $\beta$ is non-negative. The behavior in large time of the process $(V,X)$ is proved and the precise rate of convergence is pointed out by using stochastic analysis tools. To this end, we compute the moment estimates of the velocity process.

Mots clés: kinetic stochastic equation; time-inhomogeneous stochastic differential equation; Lévy process; explosion times; scaling transformations; asymptotic distributions; ergodicity; tightness

Domaines: Probabilités [math.PR]

Mihai Gradinaru, Emeline Luirard. Asymptotic behavior for a time-inhomogeneous stochastic differential equation driven by an α-stable Lévy process. 2021. ⟨hal-03478268v1⟩