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April, 2018 Reflections at infinity of time changed RBMs on a domain with Liouville branches
Zhen-Qing CHEN, Masatoshi FUKUSHIMA
J. Math. Soc. Japan 70(2): 833-852 (April, 2018). DOI: 10.2969/jmsj/07027645


Let $Z$ be the transient reflecting Brownian motion on the closure of an unbounded domain $D\subset \mathbb{R}^d$ with $N$ number of Liouville branches. We consider a diffuion $X$ on $\overline{D}$ having finite lifetime obtained from $Z$ by a time change. We show that $X$ admits only a finite number of possible symmetric conservative diffusion extensions $Y$ beyond its lifetime characterized by possible partitions of the collection of $N$ ends and we identify the family of the extended Dirichlet spaces of all $Y$ (which are independent of time change used) as subspaces of the space $\mathrm{BL}(D)$ spanned by the extended Sobolev space $H_e^1(D)$ and the approaching probabilities of $Z$ to the ends of Liouville branches.


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Zhen-Qing CHEN. Masatoshi FUKUSHIMA. "Reflections at infinity of time changed RBMs on a domain with Liouville branches." J. Math. Soc. Japan 70 (2) 833 - 852, April, 2018.


Published: April, 2018
First available in Project Euclid: 18 April 2018

zbMATH: 06902443
MathSciNet: MR3787741
Digital Object Identifier: 10.2969/jmsj/07027645

Primary: 60J50
Secondary: 31C25 , 60J65

Keywords: approaching probability , Beppo Levi space , Liouville domain , quasi-homeomorphism , Time change , transient reflecting Brownian motion , zero flux

Rights: Copyright © 2018 Mathematical Society of Japan


Vol.70 • No. 2 • April, 2018
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