Appears in collection : Non standard diffusions in fluids, kinetic equations and probability / Diffusions non standards en mécanique des fluides, équations cinétiques et probabilités

I will discuss a joint work with Jose Canizo, Cao Chuqi and Havva Yolda. I will introduce Harris's theorem which is a classical theorem from the study of Markov Processes. Then I will discuss how to use this to show convergence to equilibrium for some spatially inhomogeneous kinetic equations involving jumps including jump processes which approximate diffusion or fractional diffusion in velocity. This is the situation in which the tools of 'Hypocoercivity' are used. I will discuss the connections to hypocoercivity theory and possible advantages and disadvantages of approaches via Harris's theorem.