On unique extension of time changed reflecting Brownian motions

Let D be an unbounded domain in ℝ^d with d ≥ 3. We show that if D contains an unbounded uniform domain, then the symmetric reflecting Brownian motion (RBM) on D is transient. Next assume that RBM X on D is transient and let Y be its time change by Revuz measure 1_D(x)m(x) dx for a strictly positive continuous integrable function m on D. We further show that if there is some r > 0 so that D \ B(0, r) is an unbounded uniform domain, then Y admits one and only one symmetric diffusion that genuinely extends it and admits no killings.