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