论文标题
由$α$稳定的流程和重尾抽样驱动的超临界SDE的奇迹性
Ergodicity of supercritical SDEs driven by $α$-stable processes and heavy-tailed sampling
论文作者
论文摘要
令$α\ in（0,2）$和$ d \ in \ mathbb {n} $。考虑以下由$ \ Mathbb {r}^d $中的$α$稳定过程驱动的随机微分方程（SDE）驱动$ b：\ mathbb {r}^d \ to \ mathbb {r}^d $和$σ：\ Mathbb {r}^d \ to \ to \ Mathbb {r}^d \ otimes \ otimes \ Mathbb {r} $γ\ in（0 \ vee（1-α）^+，1] $，$ l^α_t$是$ d $ - 二维旋转不变的$α$稳定过程。在某些$ b，σ$上，我们显示$ v $ v $ - 均匀的emigruff $ p. p_ $ b，nipperative $α$稳定过程。 $（x_t（x），t \ geq 0）$。采样方案。
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.
