Functional limit theorems for processes pieced together from excursions

TL;DR

This paper introduces a new convergence concept for excursion measures and demonstrates how it ensures the convergence in law of processes constructed from excursions, with applications to various stochastic processes.

Contribution

It develops a notion of convergence for excursion measures and proves that this leads to convergence in law of the associated pieced-together processes, with multiple applications.

Findings

Convergence of excursion measures implies convergence in law of the constructed processes.

Applied to homogenization theorems for various self-similar and fractal processes.

Provides a unified framework for analyzing process convergence via excursions.

Abstract

A notion of convergence of excursion measures is introduced. It is proved that convergence of excursion measures implies convergence in law of the processes pieced together from excursions. This result is applied to obtain homogenization theorems of jumping-in extensions for positive self-similar Markov processes, for Walsh diffusions and for the Brownian motion on the Sierpi\'nski gasket.