Langevin process reflected on a partially elastic boundary II

TL;DR

This paper investigates a Langevin process reflected on a boundary with partial elasticity, showing finite bounce accumulation without absorption, and constructs a unique recurrent extension using stochastic PDEs and h-transform techniques.

Contribution

It introduces a novel recurrent extension of the reflected Langevin process with partial elasticity, and characterizes it as the unique solution to the associated stochastic PDE.

Findings

Finite bounce accumulation occurs for c < 0.1630.

The resurrected process is a recurrent extension.

The process is uniquely characterized by a stochastic PDE.

Abstract

A particle subject to a white noise external forcing moves like a Langevin process. Consider now that the particle is reflected at a boundary which restores a portion c of the incoming speed at each bounce. For c strictly smaller than the critical value 0.1630, the bounces of the reflected process accumulate in a finite time. We show that nonetheless the particle is not necessarily absorbed after this time. We define a "resurrected" reflected process as a recurrent extension of the absorbed process, and study some of its properties. We also prove that this resurrected reflected process is the unique solution to the stochastic partial differential equation describing the model. Our approach consists in defining the process conditioned on never being absorbed, via an h-transform, and then giving the Ito excursion measure of the recurrent extension thanks to a formula fairly similar to Imhof's relation.