Diffusion Approximation for a Class of Transport Processes with Physical Reflection Boundary Conditions

Abstract

Consider a stochastic process consisting of the pair of a position and a velocity, in a piecewise $\mathscr{L}^1 d$-dimensional domain. In the interior of the domain the dynamics are assigned by a potential and by random changes of the velocity occurring at exponentially distributed times, according to a probability distribution which may depend on the current position and velocity. On the boundary the process reflects physically (the angle of reflection equals the angle of incidence). First it is shown that the process is well defined for all times. Then, when the coefficients depend on a diverging parameter $N$, in particular such that the speed and the jump rate of the velocity go to $\infty$ with order $\sqrt{N}$ and at least $N$ respectively, a diffusion approximation is sought. The position process is represented as a solution of a Skorohod reflection equation: A skewing effect on the boundary results from the interaction between the dynamics and the reflection law, so that the direction of reflection is in general oblique. The assumption that the mean change of the velocity in the interior is linear in the current velocity, up to order at least $1/2$ in $1/N$, ensures that the cone of directions of reflection is independent of $N$. The continuity properties of the Skorohod oblique reflection problem enable one to show tightness of the position processes without having to estimate explicitly local times (boundary layers) and, together with a suitable law of large numbers for the velocity, allow one to identify the limit stochastic differential equation with oblique reflection. The theory is illustrated by several applications, in particular one to a mechanical model of Brownian motion.