Iterated Brownian motion in an open set

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 tau is the first exit time of iterated Brownian motion from the solid, then P(tau > 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(tau > t) for both bounded and unbounded sets. We also discuss a strange connection between iterated Brownian motion and the parabolic operator 1/8 Delta(2) - partial derivative/partial derivativet.