A new proof of an Engelbert-Schmidt type zero-one law for time-homogeneous diffusions

TL;DR

This paper presents a new, more intuitive proof of an Engelbert-Schmidt type zero-one law for time-homogeneous diffusions, providing deterministic criteria for the convergence of integral functionals without relying on complex advanced methods.

Contribution

The authors introduce a novel proof method based on stochastic time change and Feller's test, simplifying the understanding of zero-one laws for diffusions compared to previous approaches.

Findings

Provides a deterministic criterion for convergence of integral functionals

Links integral functional to explosion time of an associated diffusion

Simplifies proof by avoiding advanced theorems

Abstract

In this paper we give a new proof to an Engelbert-Schmidt type zero-one law for time-homogeneous diffusions, which provides deterministic criteria for the convergence of integral functional of diffusions. Our proof is based on a slightly stronger assumption than that in Mijatovi\'c and Urusov (2012), and utilizes stochastic time change and Feller's test of explosions. It does not rely on advanced methods such as the first Ray-Knight theorem, Wiliam's theorem, Shepp's dichotomy result for Gaussian processes or Jeulin's lemma as in the previous literature(see Mijatovi\'c and Urusov (2012) for a pointer to the literature). The new proof has an intuitive interpretation as we link the integral functional to the explosion time of an associated diffusion process.