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

Abstract

Let $\alpha \in (0,2)$ and $d\in \mathbb{N}$. Consider the following stochastic differential equation (SDE) driven by $\alpha$-stable process in ${\mathbb{R}}^d$: $$d{X}_t=b(X_t)dt+\sigma (X_{t-})d{L}_t^{\alpha },X_0=x\in {\mathbb{R}}^d,$$ where $b:{\mathbb{R}}^d\to {\mathbb{R}}^d$ and $\sigma :{\mathbb{R}}^d\to {\mathbb{R}}^d\otimes {\mathbb{R}}^d$ are locally $\gamma$-Hölder continuous with $\gamma \in (0\vee (1-\alpha {)}^{+},1]$, $L_t^{\alpha }$ is a $d$-dimensional rotationally invariant $\alpha$-stable process. Under some dissipative and non-degenerate assumptions on $b,\sigma$, we show the $V$-uniformly exponential ergodicity for the semigroup $P_t$ associated with $(X_t(x),t\ge 0)$. 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 $\alpha$ goes to zero, the diffusion coefficient $\sigma$ can grow faster than drift $b$. As applications, we put forward a new heavy-tailed sampling scheme.

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