The Doeblin Condition for a Class of Diffusions with Jumps†

We prove non-explosiveness and a lower bound of the spectral gap via the strong Doeblin condition for a large class of stochastic processes evolving in the interior of a region D ⊆ Rd with boundary ∂D according to an underlying Markov process with transition probabilities p(t, x, dy), undergoing jumps to a random point x in D with distribution νξ(dx) as soon as they reach a boundary point ξ. Besides usual regularity conditions on p(t, x, dy), we require a tightness condition on the family of measures νξ, preventing mass from escaping to the boundary. The setup can be applied to a multitude of models considered recently, including a particle system with the Bak-Sneppen dynamics from evolutionary biology.