Summary
Suppose thatD ⊂ ℝn,n≧2, is a Lipschitz domain and letN t (r) be the number of excursions of Brownian motion insideD with diameter greater thanr which started before timet. ThenrN t (r) converges asr→0 to a constant multiple of local time on ∂D, a.s. and inL p for allp<∞. The limit need not exist or may be trivial (0 or ∞) in Hölder domains, non-tangentially accessible domains and domains whose boundaries have finite surface area.
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The first author was supported by NSF Postdoctoral Fellowship
The second and third authors were supported in part by NSF grants DMS 8701073 and DMS 8702620
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Bañuelos, R., Bass, R.F. & Burdzy, K. A representation of local time for Lipschitz surfaces. Probab. Th. Rel. Fields 84, 521–547 (1990). https://doi.org/10.1007/BF01198318
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DOI: https://doi.org/10.1007/BF01198318
Keywords
- Stochastic Process
- Brownian Motion
- Probability Theory
- Local Time
- Mathematical Biology