A functional limit theorem for additive functionals

TL;DR

This paper establishes a general limit theorem for additive functionals of diffusions converging to Lévy subordinators, with applications in neuroscience models involving reflected diffusions.

Contribution

It provides explicit conditions for convergence and characterizes the law of the limit, including a new limiting regime for Wright-Fisher and Feller diffusions.

Findings

Identified a novel limiting regime for certain diffusions.

Provided explicit sufficient conditions for convergence.

Applied results to neuroscience models involving reflected diffusions.

Abstract

We study a general limiting framework for the convergence of sequences of additive functionals of diffusions to L\'evy subordinators, and provide explicit sufficient conditions that both ensure convergence and characterize the law of the limit. As an application, we identify a novel limiting regime for Wright-Fisher and Feller diffusions in the reflecting case and describe the corresponding limiting subordinator. This work is motivated by, and has applications in, neuroscience, where reflected diffusions are used to parametrize synchrony in doubly-stochastic models of spiking activity.