Coupling approach for exponential ergodicity of stochastic Hamiltonian systems with Lévy noises

Abstract

We establish exponential ergodicity for the stochastic Hamiltonian system (Xt,Vt)t≥0 on R2d with Lévy noises dXt=(aXt+bVt)dt,dVt=U(Xt,Vt)dt+dLt,where a≥0, b>0, U:R2d→Rd and (Lt)t≥0 is an Rd-valued pure jump Lévy process. The approach is based on a new refined basic coupling for Lévy processes and a Lyapunov function for stochastic Hamiltonian systems. In particular, we can handle the case that U(x,v)=−v−∇U0(x) with double well potential U0 which is super-linear growth at infinity such as U0(x)=c1(1+|x|2)l−c2|x|2 with l>1 or U0(x)=c1e(1+|x|2)l−c2|x|2 with l>0 for any c1,c2>0, and also deal with the case that the Lévy measure ν of (Lt)t≥0 is degenerate in the sense that ν(dz)≥c|z|d+θ01{0<z1≤1}dzfor some c>0 and θ0∈(0,2), where z1 is the first component of the vector z∈Rd.