Convergence of Integral Functionals of One-Dimensional Diffusions

TL;DR

This paper investigates the conditions under which integral functionals of one-dimensional diffusion processes converge, providing deterministic characterizations and two different proofs using Brownian motion path properties.

Contribution

It offers a new deterministic characterization of convergence for integral functionals of diffusions, utilizing two distinct approaches based on Brownian motion path properties.

Findings

Provides necessary and sufficient conditions for convergence.

Uses Williams theorem and Ray-Knight theorem in proofs.

Connects diffusion integral convergence to Brownian motion path properties.

Abstract

In this expository paper we describe the pathwise behaviour of the integral functional $\int_0^t f(Y_u)\,\dd u$ for any $t\in[0,\zeta]$, where $\zeta$ is (a possibly infinite) exit time of a one-dimensional diffusion process $Y$ from its state space, $f$ is a nonnegative Borel measurable function and the coefficients of the SDE solved by $Y$ are only required to satisfy weak local integrability conditions. Two proofs of the deterministic characterisation of the convergence of such functionals are given: the problem is reduced in two different ways to certain path properties of Brownian motion where either the Williams theorem and the theory of Bessel processes or the first Ray-Knight theorem can be applied to prove the characterisation.