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Let $\alpha\in(0,2)$ and $d\in\mathbb{N}$. Consider the following stochasticdifferential equation (SDE) driven by $\alpha$-stable process in$\mathbb{R}^d$: $$ dX_t=b(X_t)dt+\sigma(X_{t-})d L^{\alpha}_t, \quadX_0=x\in\mathbb{R}^d, $$ where $b:\mathbb{R}^d\to\mathbb{R}^d$ and$\sigma:\mathbb{R}^d\to\mathbb{R}^d\otimes\mathbb{R}^d$ are locally$\gamma$-H\"older continuous with $\gamma\in(0\vee(1-\alpha)^+,1]$,$L^\alpha_t$ is a $d$-dimensional rotationally invariant $\alpha$-stableprocess. Under some dissipative and non-degenerate assumptions on $b,\sigma$,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 heatkernel estimates recently established in \cite{MZ20} through showing the strongFeller property and the irreducibility of $P_t$. It is interesting that when$\alpha$ goes to zero, the diffusion coefficient $\sigma$ can grow faster thandrift $b$. As applications, we put forward a new heavy-tailed sampling scheme.

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