Ergodicity for multidimensional jump diffusions with position dependent jump rate

We consider a jump type diffusion X = (Xt) with infinitesimal generator given by Lψ(x) = 1/2 ∑ a ij (x) ∂2 ψ(x)/∂x i ∂x j + g(x)∇ψ(x) + ∫ (ψ(x + c(z, x)) − ψ(x))γ(z, x)µ(dz) where µ is of infinite total mass. We prove Harris recurrence of X using a regeneration scheme which is entirely based on the jumps of the process. Moreover we state explicit conditions in terms of the coefficients of the process allowing to control the speed of convergence to equilibrium in terms of deviation inequalities for integrable additive functionals.