Langevin process reflected on a partially elastic boundary I

Emmanuel Jacob (1)

1 LPMA - Laboratoire de Probabilités et Modèles Aléatoires

Consider a Langevin process, that is an integrated Brownian motion, constrained to stay in $[0,\infty)$ by a partially elastic boundary at 0. If the elasticity coefficient of the boundary is greater than or equal to $\exp(-\sqrt \pi /3)$, bounces will not accumulate in a finite time when the process starts from the origin with strictly positive velocity. We will endeavor to 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.

Mots clés: Langevin process; Second order reflexion; Renewal theory; Stationarity; Ladder height processes

Domaines: Probabilités [math.PR]

Emmanuel Jacob. Langevin process reflected on a partially elastic boundary I. 2010. ⟨hal-00472601v2⟩