Iterated Brownian motion in an open set

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 τis 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}Δ^2-\frac{\partial}{\partial t}.

References in corpus (2)

• Brownian-Time Processes: The PDE Connection and the Half-Derivative Generator
• Brownian-Time Processes: The PDE Connection II and the Corresponding Feynman-Kac Formula

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