Ergodicity of supercritical SDEs driven by $α$-stable processes and heavy-tailed sampling

Let $α\in(0,2)$ and $d\in\mathbb{N}$. Consider the following stochastic differential equation (SDE) driven by $α$-stable process in $\mathbb{R}^d$: $dX_t=b(X_t)dt+σ(X_{t-})d L^α_t, \quad X_0=x\in\mathbb{R}^d,$ where $b:\mathbb{R}^d\to\mathbb{R}^d$ and $σ:\mathbb{R}^d\to\mathbb{R}^d\otimes\mathbb{R}^d$ are locally $γ$-Hölder continuous with $γ\in(0\vee(1-α)^+,1]$, $L^α_t$ is a $d$-dimensional rotationally invariant $α$-stable process. Under some dissipative and non-degenerate assumptions on $b,σ$, we show the $V$-uniformly exponential ergodicity for the semigroup $P_t$ associated with $(X_t(x),t\geq 0)$. Our proofs are mainly based on the heat kernel estimates recently established in \cite{MZ20} through showing the strong Feller property and the irreducibility of $P_t$. It is interesting that when $α$ goes to zero, the diffusion coefficient $σ$ can grow faster than drift $b$. As applications, we put forward a new heavy-tailed sampling scheme.

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