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

Zhen Qing Chen, Masatoshi Fukushima

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Abstract

Let Z be the transient reflecting Brownian motion on the closure of an unbounded domain D ⊂ ℝd with N number of Liouville branches. We consider a diffuion X on 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 BL(D) spanned by the extended Sobolev space He1(D) and the approaching probabilities of Z to the ends of Liouville branches.

Original languageEnglish
Pages (from-to)833-852
Number of pages20
JournalJournal of the Mathematical Society of Japan
Volume70
Issue number2
DOIs
Publication statusPublished - 2018
Externally publishedYes

Keywords

  • Approaching probability
  • Beppo Levi space
  • Liouville domain
  • Quasi-homeomorphism
  • Time change
  • Transient reflecting Brownian motion
  • Zero flux

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Chen, Z. Q., & Fukushima, M. (2018). Reflections at infinity of time changed RBMs on a domain with Liouville branches. Journal of the Mathematical Society of Japan, 70(2), 833-852. https://doi.org/10.2969/jmsj/07027645