Diffusion approximation for a simple kinetic model with asymmetric interface

TL;DR

This paper analyzes a stochastic particle model with asymmetric interface interactions, demonstrating that its diffusion limit is either a minimal or skew Brownian motion depending on killing probabilities.

Contribution

It introduces a diffusion approximation for a kinetic model with asymmetric interface interactions, revealing the limiting process as either minimal or skew Brownian motion.

Findings

Limit process is minimal Brownian motion with positive killing probability.

Limit process is skew Brownian motion when there is no killing.

The model captures asymmetric interface effects in stochastic particle motion.

Abstract

We study a diffusion approximation for a model of stochastic motion of a particle in one spatial dimension. The velocity of the particle is constant but the direction of the motion undergoes random changes with a Poisson clock. Moreover, the particle interacts with an interface in such a way that it can randomly be reflected, transmitted, or killed, and the corresponding probabilities depend on whether the particle arrives at the interface from the left, or right. We prove that the limit process is a minimal Brownian motion, if the probability of killing is positive. In the case of no killing, the limit is a skew Brownian motion.