Stopping Time Convergence for Processes Associated with Dirichlet Forms

Convergence is proved for solutions un of Dirichlet problems in regions with many small excluded sets (holes), as the holes become smaller and more numerous. The problem is formulated probabilistically in the context of general Dirichlet forms, for random and nonrandom excluded sets. Sufficient conditions are given in Theorem 2.1 under which the sequence of entrance times or hitting times of the excluded sets converges in the stable topology. Convergence in the stable topology is a strengthened form of convergence in distribution, introduced by Rényi. Stable convergence of the entrance times implies convergence of the solutions un of the corresponding Dirichlet problems. Theorem 2.1 applies to Dirichlet forms such that the Markov process associated with the form has continuous paths and satisfies an absolute continuity condition for occupation time measures (Eq. 2.4). Conditions for convergence are formulated in terms of the sum of the expectations of the equilibrium measures for the excluded sets. The proof of convergence uses the fact that any martingale with respect to the natural filtration of the process must be continuous. In the case that the excluded sets are iid random, Theorem 2.1 strengthens previous results for the classical Brownian motion setting.