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Langevin process reflected on a partially elastic boundary II

By Emmanuel Jacob

Abstract

27 pages, 1 figure. This article follows naturally the article "Langevin process reflected on a partially elastic boundary I", though it can be read separatelyA 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

Topics: Langevin process, second order reflection, recurrent extension, excursion measure, stochastic partial differential equation, h-transform, MSC2010: 60J50, 60H15, [ MATH.MATH-PR ] Mathematics [math]/Probability [math.PR]
Publisher: HAL CCSD
Year: 2011
OAI identifier: oai:HAL:hal-00577024v1
Provided by: Hal-Diderot
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