A note on a.s. finiteness of perpetual integral functionals of diffusions

TL;DR

This paper establishes a criterion for the convergence of perpetual integral functionals of transient diffusions using boundary classification, aligning with known results for Bessel processes and enhancing understanding of their long-term behavior.

Contribution

It provides a new criterion for the convergence of perpetual integral functionals of transient diffusions based on boundary classification, connecting with existing results for Bessel processes.

Findings

Derived a convergence criterion for perpetual integral functionals of transient diffusions.

Confirmed the criterion's consistency with Jeulin's convergence lemma for Bessel processes.

Enhanced understanding of the boundary behavior influencing the convergence of diffusions.

Abstract

In this note, with the help of the boundary classification of diffusions, we derive a criterion of the convergence of perpetual integral functionals of transient real-valued diffusions. In the particular case of transient Bessel processes, we note that this criterion agrees with the one obtained via Jeulin's convergence lemma.