Kinetic time-inhomogeneous L{\'e}vy-driven model

TL;DR

This paper analyzes a one-dimensional kinetic stochastic model driven by a Lévy process with a time-inhomogeneous drift, establishing its long-term behavior and convergence rates through stochastic analysis.

Contribution

It introduces a new kinetic Lévy-driven model with non-linear time-inhomogeneous drift and characterizes its large-time behavior and convergence rates.

Findings

Established the large-time behavior of the process

Derived the rate of convergence using stochastic analysis

Computed moment estimates of the velocity process

Abstract

We study a one-dimensional kinetic stochastic model driven by a L{\'e}vy 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 process can be a stable L{\'e}vy process of index $\alpha$ or a general L{\'e}vy 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.