Coupling approach for exponential ergodicity of stochastic Hamiltonian systems with L\'evy noises

We establish exponential ergodicity for the stochastic Hamiltonian system $(X_t, V_t)_{t\ge0}$ on $\mathbb{R}^{2d}$ with L\'evy 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 $(L_t)_{t\ge0}$ is an $\mathbb{R}^d$-valued pure jump L\'{e}vy process. The approach is based on a new refined basic coupling for L\'evy 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 L\'evy measure $\nu$ of $(L_t)_{t\ge0}$ 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 $z\in \mathbb{R}^d$.