Ratio limit theorems for a Brownian motion killed at the boundary of a Benedicks domain. (English) Zbl 0965.60076

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 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.


60J65 Brownian motion
35B40 Asymptotic behavior of solutions to PDEs
35K05 Heat equation


Zbl 0455.31009
Full Text: DOI


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