Ratio limit theorems for a Brownian motion killed at the boundary of a Benedicks domain
Let be \(n\geq 0\) and consider \(E\), a closed proper subset of \(\mathbb{R}^{n-1}\). Also let \({\mathcal P}_E\) be the conc of positive harmonic functions in \({\mathcal D}=\mathbb{R}^n\setminus E\times \{0\}\) vanishing at \(E\). A result of \textit{M. Benedicks} [Ark. Math. 18, 53-72 (1980; Zbl 0455.31009)] states that either all functions in \({\mathcal P}_E\) are proportional or \({\mathcal P}_E\) is generated by two linearly independent, minimal positive harmonic functions. The authors characterize for an integral test both cases and remark that in both cases there is a unique, up to a multiplicative constant, positive harmonic and symmetric function. The authors show ratio limit theorems for the associated heat kernlel. When the hole is compact, the Martin boundary is two-dimensional. Sharp estimates on the life time probabilities are also obtained and the various constants appearing in the theory are identified in probabilistic terms.
scientific article; zbMATH DE number 1462547