Thomas Onskog: Existence of pathwise unique Langevin processes on polytopes with perfect reflection at the boundary

A common way to obtain Lagrangian stochastic models for turbulent diffusion is to let the particle velocity be modelled by some suitable stochastic differential equation (and to let the particle position be the time integral of the velocity). The joint position and velocity process (X,U) is then known as a Langevin process. If the position domain is bounded, some boundary conditions have to be imposed on the velocity process to ensure that the position process stays within the given domain. There has been a recent breakthrough concerning the existence of unique weak Langevin processes in the entire space, in half spaces and in domains with smooth boundary (Bossy et al 2011). In the special case when there is no drift in the stochastic differential equation for the velocity, we show the existence of pathwise unique strong Langevin processes on polytopes. The idea of the proof is to exploit an explicit projection from the real line into an interval to prove existence of pathwise unique one-dimensional Langevin processes confined to intervals and then to generalize this result to polytopes by constructing a recursion of projections.