Journal of the Mathematical Society of Japan

Reflections at infinity of time changed RBMs on a domain with Liouville branches

Zhen-Qing CHEN and Masatoshi FUKUSHIMA

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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.

Article information

J. Math. Soc. Japan, Volume 70, Number 2 (2018), 833-852.

First available in Project Euclid: 18 April 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J50: Boundary theory
Secondary: 60J65: Brownian motion [See also 58J65] 31C25: Dirichlet spaces

transient reflecting Brownian motion time change Liouville domain Beppo Levi space approaching probability quasi-homeomorphism zero flux


CHEN, Zhen-Qing; FUKUSHIMA, Masatoshi. Reflections at infinity of time changed RBMs on a domain with Liouville branches. J. Math. Soc. Japan 70 (2018), no. 2, 833--852. doi:10.2969/jmsj/07027645.

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