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

Jianhai Bao and Jian Wang

Stochastic Processes and their Applications, 2022, vol. 146, issue C, 114-142

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{00 and θ0∈(0,2), where z1 is the first component of the vector z∈Rd.

Keywords: Stochastic Hamiltonian system; Langevin dynamic; Lévy process; Refined basic coupling; Exponential ergodicity (search for similar items in EconPapers)
Date: 2022
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Citations: View citations in EconPapers (1)

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DOI: 10.1016/j.spa.2021.12.014

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