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

Abstract

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.