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Collet, Pierre; Martínez, Servet; San Martín, Jaime

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

Ann. Probab. 27, No. 3, 1160-1182 (1999).
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.
Reviewer: G.G.Vrănceanu (Bucureşti)

Cited in 5 Documents

MSC:

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

Keywords:

Brownian motion; heat kernel; ratio limit theorems; Benedicks domain

Citations:

Zbl 0455.31009
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\textit{P. Collet} et al., Ann. Probab. 27, No. 3, 1160--1182 (1999; Zbl 0965.60076)
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References:

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