Langevin Process Reflected on a Partially Elastic Boundary II

A particle subject to a white noise external forcing moves according to 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 \(c_{\mathit{crit}} =\exp (-\pi /\sqrt{3})\), the bounces of the reflected process accumulate in a finite time, yielding a very different behavior from the most studied cases of perfectly elastic reflection—c = 1—and totally inelastic reflection—c = 0. We show that nonetheless the particle is not necessarily absorbed after this accumulation of bounces. 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, for which well-posedness is nothing obvious. Our approach consists in defining the process conditioned on never being absorbed, via an h-transform, and then giving the Itō excursion measure of the recurrent extension thanks to a formula fairly similar to Imhof’s relation.