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-\frac{\partial}{\partial t}

We describe a connection between the semigroup of a symmetric stable process with rational index and higher order partial differential equations. As an application, we obtain a variational formula for the eigenvalues associated with the process killed upon leaving a bounded open set D. The variational formula is more "user friendly" than the classical Rayleigh-Ritz formula. We illustrate this by obtaining upper bounds on the eigenvalues in terms of Dirichlet eigenvalues of the Laplacian on D. These results generalize some work of Bañuelos and Kulczycki on the Cauchy process. Along the way we prove an operator inequality for the operators associated with the transition densities of Brownian motion and the Brownian motion killed upon leaving D.

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