# Stopping time convergence for processes associated with Dirichlet forms

## TL;DR

This paper proves convergence of solutions to Dirichlet problems involving Markov processes with many small holes, under certain conditions, using stable topology, which strengthens traditional convergence notions.

## Contribution

It establishes stable convergence of entrance and hitting times for processes associated with general Dirichlet forms in complex regions with small excluded sets.

## Key findings

- Convergence of solutions in regions with many small holes
- Stable topology used for stronger convergence results
- Additional results on random center models

## Abstract

Convergence is proved for solutions of Dirichlet problems in regions with many small excluded sets (holes), as the holes become smaller and more numerous. The problem is formulated in the context of Markov processes associated with general Dirichlet forms, for random and nonrandom excluded sets. Sufficient conditions are given 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 Renyi. Stable convergence of the entrance times implies convergence of the solutions of the corresponding Dirichlet problems. Some additional results are given in a supplement on random center models.