Iterated Brownian motion in an open set

2004/07/08 by R. Dante DeBlassie · 4 citations
Economics, Econometrics and Finance · Mathematics · #Mathematical Dynamics and Fractals #Stochastic processes and financial applications #Stochastic processes and statistical mechanics #math.PR #msc:60J65 #msc:60K99

paper · pdf · doi:10.1214/105051604000000404

published as Annals of Probability 2004, Vol. 14, No. 3, 1529-1558

arxiv created 2004/07/08 · openalex publication_date 2004/07/14 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Suppose a solid has a crack filled with a gas. If the crack reaches the surrounding medium, how long does it take the gas to diffuse out of the crack? Iterated Brownian motion serves as a model for diffusion in a crack. If τ is the first exit time of iterated Brownian motion from the solid, then P(τ>t) can be viewed as a measurement of the amount of contaminant left in the crack at time t. We determine the large time asymptotics of P(τ>t) for both bounded and unbounded sets. We also discuss a strange connection between iterated Brownian motion and the parabolic operator (1)/(8)Δ²-(∂)/(∂ t).

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