Two-sided estimates for the transition densities of symmetric Markov processes dominated by stable-like processes in open sets

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• 作者：Kyung-Youn Kim^a ; ^{eunicekim@snu.ac.kr" class="auth_mail} ; Panki Kim^a ; ^b ; ^{pkim@snu.ac.kr" class="auth_mail}
• 关键词：primary ; 60J35 ; 47G20 ; 60J75 ; secondary ; 47D07
• 刊名：Stochastic Processes and their Applications
• 出版年：September 2014
• 年：2014
• 卷：124
• 期：9
• 页码：3055-3083
• 全文大小：350 K

文摘

In this paper, we study sharp Dirichlet heat kernel estimates for a large class of symmetric Markov processes in C^1,畏$C^{1,畏}$ open sets. The processes are symmetric pure jump Markov processes with jumping intensity 魏(x,y)蠄₁(|x−y|)⁻¹|x−y|^−d−伪$魏(x,y){蠄}_1{\left(|x-y|\right)}^{-1}{|x-y|}^{-d-伪}$, where 伪∈(0,2)$伪\in (0,2)$. Here, 蠄₁${蠄}_1$ is an increasing function on [0,∞)$[0,\infty )$, with 蠄₁(r)=1${蠄}_1\left(r\right)=1$ on 0<r≤1$0<r\le 1$ and c₁e^c₂r尾≤蠄₁(r)≤c₃e^c₄r尾$c_1{e}^{c_2{r}^{尾}}\le {蠄}_1\left(r\right)\le c_3{e}^{c_4{r}^{尾}}$ on r>1$r>1$ for 尾∈[0,∞]$尾\in [0,\infty ]$, and 魏(x,y)$魏(x,y)$ is a symmetric function confined between two positive constants, with |魏(x,y)−魏(x,x)|≤c₅|x−y|^蟻$|魏(x,y)-魏(x,x)|\le c_5{|x-y|}^{蟻}$ for 26b050142c9" title="Click to view the MathML source">|x−y|<1$|x-y|<1$ and 蟻>伪/2$蟻>伪/2$. We establish two-sided estimates for the transition densities of such processes in C^1,畏$C^{1,畏}$ open sets when 畏∈(伪/2,1]$畏\in (伪/2,1]$. In particular, our result includes (relativistic) symmetric stable processes and finite-range stable processes in C^1,畏$C^{1,畏}$ open sets when 畏∈(伪/2,1]$畏\in (伪/2,1]$.