Scaling limit of a kinetic inhomogeneous stochastic system in the quadratic potential

TL;DR

This paper analyzes the long-time behavior of a particle in a quadratic potential influenced by time-inhomogeneous friction and stable Lévy noise, identifying three distinct regimes based on system parameters.

Contribution

It characterizes the asymptotic scaling limits of the particle's velocity and position under complex stochastic dynamics with inhomogeneous friction and Lévy noise.

Findings

Three regimes for long-time behavior depending on friction and noise stability index

Identification of scaling limits for velocity and position

Analysis of the interplay between frictional decay and Lévy noise stability

Abstract

We consider a particle evolving in the quadratic potential and subject to a time-inhomogeneous frictional force and to a random force. The couple of its velocity and position is solution to a stochastic differential equation driven by an $\alpha$-stable L{\'e}vy process with $\alpha \in (1,2]$ and the frictional force is of the form $t^{-\beta}\text{sgn}(v)|v|^\gamma$. We identify three regimes for the behavior in long-time of the couple velocity-position with a suitable rescaling, depending on the balance between the frictional force and the index of stability $\alpha$ of the noise.