arXiv 1002.11212010-02-05math.PRmath.AP

$C^{1,1}$开集上$Δ+Δ^{α/2}$的热核估计

Heat Kernel Estimate for $Δ+Δ^{α/2}$ in $C^{1,1}$ open sets

Zhen-Qing Chen, Panki Kim, Renming Song

摘要：

考虑$\bR^d$上的一族伪微分算子$\{Δ+ a^αΔ^{α/2}; a\in (0, 1]\}$，对于每个从$Δ$到$Δ+ Δ^{α/2}$连续演化的$d\geq 1$，其中$α\in (0, 2)$ .它在$\bR^d$中产生了一族Lévy过程$\{X^a, a\in (0, 1]\}$，其中$X^a$是布朗运动和一个独立的对称$α$ -稳定过程的和。在一类开集（包括有界的$C^{1, 1}$（可能不连通的）开集）中，建立了具有零外部条件的$Δ+ a^α Δ^{α/2}$的热核的双边估计.这个热核也是布朗运动和一个独立的对称$α$ -稳定过程的重量$a$在这样的开集的总和的转移密度。我们的结果是第一个尖锐的双边估计的马尔可夫过程的转移密度的扩散和跳跃组件在开集。此外，我们的结果在$a$中是一致的，因为估计中的常数与$a\in (0, 1]$无关，所以它恢复了布朗运动的Dirichlet热核估计.在时间$t$上积分热核估计，我们恢复了$\bR^d$中有界$C^{1,1}$开集上$X^a$的双边尖锐一致格林函数估计，这是最近在\cite{CKSV2}中通过使用完全不同的方法建立的.

英文摘要：

We consider a family of pseudo differential operators $\{Δ+ a^αΔ^{α/2}; a\in (0, 1]\}$ on $\bR^d$ for every $d\geq 1$ that evolves continuously from $Δ$ to $Δ+ Δ^{α/2}$, where $α\in (0, 2)$. It gives rise to a family of Lévy processes $\{X^a, a\in (0, 1]\}$ in $\bR^d$, where $X^a$ is the sum of a Brownian motion and an independent symmetric $α$-stable process with weight $a$. We establish sharp two-sided estimates for the heat kernel of $Δ+ a^α Δ^{α/2}$ with zero exterior condition in a family of open subsets, including bounded $C^{1, 1}$ (possibly disconnected) open sets. This heat kernel is also the transition density of the sum of a Brownian motion and an independent symmetric $α$-stable process with weight $a$ in such open sets. Our result is the first sharp two-sided estimates for the transition density of a Markov process with both diffusion and jump components in open sets. Moreover, our result is uniform in $a$ in the sense that the constants in the estimates are independent of $a\in (0, 1]$ so that it recovers the Dirichlet heat kernel estimates for Brownian motion by taking $a\to 0$. Integrating the heat kernel estimates in time $t$, we recover the two-sided sharp uniform Green function estimates of $X^a$ in bounded $C^{1,1}$ open sets in $\bR^d$, which were recently established in \cite{CKSV2} by using a completely different approach.

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摘要：

英文摘要：

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