We use jump processes with stochastic intensity to construct a class of reflection laws for billiard processes in the unit interval whose stationary distribution for the billiard position and its velocity is the product of the uniform distribution and the standard normal distribution. These billiard processes have Markovian reflection laws, meaning their velocity is constant between reflections but changes in a Markovian way at reflection times.
"Billiards with Markovian reflection laws." Electron. J. Probab. 24 1 - 32, 2019. https://doi.org/10.1214/19-EJP398