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.