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 language | English |
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Pages (from-to) | 833-852 |
Number of pages | 20 |
Journal | Journal of the Mathematical Society of Japan |
Volume | 70 |
Issue number | 2 |
DOIs | |
Publication status | Published - 2018 |
Externally published | Yes |
Keywords
- Approaching probability
- Beppo Levi space
- Liouville domain
- Quasi-homeomorphism
- Time change
- Transient reflecting Brownian motion
- Zero flux