Langevin process reflected on a partially elastic boundary I

TL;DR

This paper studies a Langevin process with a partially elastic boundary, showing conditions for non-accumulation of bounces and establishing a unique entrance law from the boundary, contrasting with deterministic reflection cases.

Contribution

It proves the existence and uniqueness of an entrance law for the Langevin process with a partially elastic boundary at a critical elasticity coefficient.

Findings

Bounces do not accumulate in finite time if elasticity ≥ 0.16.

Unique entrance law exists from the boundary with zero velocity.

Method uses properties of real-valued random walks and spatial stationarity.

Abstract

Consider a Langevin process, that is an integrated Brownian motion, constrained to stay on the nonnegative half-line by a partially elastic boundary at 0. If the elasticity coefficient of the boundary is greater than or equal to a critical value (0.16), bounces will not accumulate in a finite time when the process starts from the origin with strictly positive velocity. We will show that there exists then a unique entrance law from the boundary with zero velocity, despite the immediate accumulation of bounces. This result of uniqueness is in sharp contrast with the literature on deterministic second order reflection. Our approach uses certain properties of real-valued random walks and a notion of spatial stationarity which may be of independent interest.