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

Let Z be the transient reflecting Brownian motion on the closure of an unbounded domain D ⊂ ℝ^d 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 H_e¹(D) and the approaching probabilities of Z to the ends of Liouville branches.