Reflecting a Langevin Process at an Absorbing Boundary

Langevin [8] introduced a probabilistic model to describe the evolution of a particle under random external forcing. Langevin’s motion has smooth trajectories, in the sense that if Yt stands for the position of the particle at time t, then the velocity Ẏt := dYt/dt is well-defined and finite everywhere. We assume that the external force (i.e. the derivative of the velocity) is a white noise, so that Ẏt = Ẏ0 +Wt , where W = (Wt, t ≥ 0) is a standard Wiener process , and Yt can then be expressed in terms of the integral of the latter. Plainly the Langevin process Y = (Yt, t ≥ 0) is not Markovian, however the pair (Y, Ẏ ), which is often called Kolmogorov’s process, is Markov. We refer to Lachal [7] for an interesting introduction to this field, historical comments, and a long list of references. Recently, Maury [9] has considered the situation when the particle may hit an obstacle and