We consider a Brownian motion in a Benedicks domain with absorption at the boundary. We show ratio limit theorems for the associated heat kernel. When the hole is compact, therefore the Martin boundary is two dimensional; we obtain sharp estimates on the lifetime probabilities and we identify, in probabilistic terms, the various constants appearing in the theory.
"Ratio Limit Theorems for a Brownian Motion Killed at the Boundary of a Benedicks Domain." Ann. Probab. 27 (3) 1160 - 1182, July 1999. https://doi.org/10.1214/aop/1022677443