Diffusion approximation for a simple kinetic model with asymmetric interface

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. Moreover, we construct a cosine family related to the skew Brownian motion and provide a new derivation of transition probability densities for this process.

Automated summary

This paper studies a diffusion approximation for a model of stochastic motion of a particle in one spatial dimension. The direction of the particle's motion undergoes random changes with a Poisson clock, and it interacts with an interface that can reflect, transmit, or kill the particle with probabilities depending on its arrival direction. The study proves that if the probability of killing is positive, the limit process is a minimal Brownian motion. In the case of no killing, the limit is a skew Brownian motion. The paper also introduces a cosine family related to the skew Brownian motion and provides a new derivation of transition probability densities for this process.