  Fractional Cauchy problems replace the usual first-order time derivative by a
fractional derivative. This paper develops classical solutions and stochastic
analogues for fractional Cauchy problems in a bounded domain
$D\subset\mathbb{R}^d$ with Dirichlet boundary conditions. Stochastic solutions
are constructed via an inverse stable subordinator whose scaling index
corresponds to the order of the fractional time derivative. Dirichlet problems
corresponding to iterated Brownian motion in a bounded domain are then solved
by establishing a correspondence with the case of a half-derivative in time.

Probability;Analysis of PDEs