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

Abstract

We establish exponential ergodicity for the stochastic Hamiltonian system $(X_t,V_t{)}_{t\ge 0}$ on ${\mathbb{R}}^{2d}$ with Lévy noises \begin{align*} \begin{cases} \mathrm{d} X_t=\big(a X_t+bV_t\big)\,\mathrm{d} t,\\ \mathrm{d} V_t=U(X_t,V_t)\,\mathrm{d} t+\mathrm{d} L_t, \end{cases} \end{align*} where $a\ge 0$, $b>0$, $U:{\mathbb{R}}^{2d}\to {\mathbb{R}}^d$ and $(L_t{)}_{t\ge 0}$ is an ${\mathbb{R}}^d$-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 $U(x,v)=-v-\nabla U_0(x)$ with double well potential $U_0$ which is super-linear growth at infinity such as $U_0(x)=c_1(1+|x{|}^2{)}^l-c_2|x{|}^2$ with $l>1$ or $U_0(x)=c_1{\mathrm{e}}^{(1+|x{|}^2{)}^l}-c_2|x{|}^2$ with $l>0$ for any $c_1,c_2>0$, and also deal with the case that the Lévy measure $\nu$ of $(L_t{)}_{t\ge 0}$ is degenerate in the sense that $$\nu (\mathrm{d}z)\ge \frac{c}{|z{|}^{d+{\theta }_0}}{\mathbb{I}}_{\{0<z_1\le 1\}}\mathrm{d}z$$ for some $c>0$ and ${\theta }_0\in (0,2)$, where $z_1$ is the first component of the vector $z\in {\mathbb{R}}^d$.

22 pages

References in corpus (2)

• Coupling and exponential ergodicity for stochastic differential equations driven by Lévy processes
• Exponential ergodicity for SDEs and McKean-Vlasov processes with Lévy noise