Continuous-Space Markov Processes with Jumps.

Summary From now on, Markov processes with continuous state space (${\mathbb{R}}^d$ for some or one of its closed subsets) are considered. Their rigorous study requires advanced measure-theoretic tools, but we limit ourselves to developing the reader’s intuition, notably by pathwise constructions leading to simulations. We first emphasize the strong similarity between such Markov processes with constant trajectories between isolated jumps and discrete space ones. We then introduce Markov processes with sample paths following an ordinary differential equation between isolated jumps. In both cases, the Kolmogorov equations and Feynman–Kac formula are established. This is applied to kinetic equations coming from statistical Mechanics. These describe the time evolution of the instantaneous distribution of particles in phase space (position-velocity), when the particle velocity jumps at random instants in function of the particle position and velocity.