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.
Pierre Collet. Servet Martínez. Jaime San Martín. "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