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arXiv 1501.04595math.PR
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多锥域热核的渐近性

Asymptotics for the heat kernel in multicone domains

Pierre Collet, Mauricio Duarte, Servet Martinez, Arturo Prat-Waldron, Jaime San Martin

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31 pages
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摘要：

多锥域$Ω\subseteq \mathbb{R}^n$是一个开的、连通集，类似于远离原点的锥的有限集合。我们研究了衰变率的时间的热核$p(t,x,y)$的布朗运动后退出$Ω$，使用概率和分析技术。我们发现，衰减是多项式，我们的特点$\lim_{t\to\infty} t^{1+α}p(t,x,y)$的马丁边界的$Ω$在无穷远，其中$α>0$依赖于几何的$Ω$。接下来我们用$κ= 1+α- n/2$导出$t^{κ/2}\mathbb{P}_x(T >t)$的类似结果，其中$T$是$Ω$的退出时间。最后，我们推导出重正化Yaglom极限的生存条件下的过程。

英文摘要：

A multi cone domain $Ω\subseteq \mathbb{R}^n$ is an open, connected set that resembles a finite collection of cones far away from the origin. We study the rate of decay in time of the heat kernel $p(t,x,y)$ of a Brownian motion killed upon exiting $Ω$, using both probabilistic and analytical techniques. We find that the decay is polynomial and we characterize $\lim_{t\to\infty} t^{1+α}p(t,x,y)$ in terms of the Martin boundary of $Ω$ at infinity, where $α>0$ depends on the geometry of $Ω$. We next derive an analogous result for $t^{κ/2}\mathbb{P}_x(T >t)$, with $κ= 1+α- n/2$, where $T$ is the exit time form $Ω$. Lastly, we deduce the renormalized Yaglom limit for the process conditioned on survival.

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