Asymptotics for the heat kernel in multicone domains

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• 作者：Pierre Collet^a ; ^{pierre.collet@cpht.polytechnique.fr" class="auth_mail" title="E-mail the corresponding author} ; Mauricio Duarte^b ; ¹ ; ^{mauricio.duarte@unab.cl" class="auth_mail" title="E-mail the corresponding author} ; Servet Mart&iacute ; nez^c ; ² ; ^{smartine@dim.uchile.cl" class="auth_mail" title="E-mail the corresponding author} ; Arturo Prat-Waldron^d ; ^{arturo@mpim-bonn.mpg.de" class="auth_mail" title="E-mail the corresponding author} ; Jaime San Mart&iacute ; n^c ; ² ; ^{jsanmart@dim.uchile.cl" class="auth_mail" title="E-mail the corresponding author}
• 关键词：primary ; 60J65 ; 35K08 ; 35B40 ; secondary ; 60H30
• 刊名：Journal of Functional Analysis
• 出版年：2016
• 出版时间：15 February 2016
• 年：2016
• 卷：270
• 期：4
• 页码：1269-1298
• 全文大小：509 K

文摘

A multicone domain Ω⊆Rⁿ$\mathrm{\Omega }\subseteq {\mathbb{R}}^n$ 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)$p(t,x,y)$ 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→∞⁡t^1+αp(t,x,y)${\lim }_{t\to \infty }{t}^{1+\alpha }p(t,x,y)$ in terms of the Martin boundary of Ω at infinity, where α>0$\alpha >0$ depends on the geometry of Ω. We next derive an analogous result for t^κ/2P_x(T>t)$t^{\kappa /2}{\mathbb{P}}_x(T>t)$, with κ=1+α−n/2$\kappa =1+\alpha -n/2$, where T is the exit time from Ω. Lastly, we deduce the renormalized Yaglom limit for the process conditioned on survival.