A New Approach to Inverse Local Times

TL;DR

This paper provides a new, more accessible proof of Knight's theorem on Levy measures of gap diffusions, replacing complex spectral analysis with alternative methods.

Contribution

It introduces a novel proof of Knight's theorem on Levy measures of gap diffusions, avoiding extensive spectral Krein Theory techniques.

Findings

New proof of Knight's theorem presented

Simplifies understanding of Levy measures for gap diffusions

Enhances mathematical tools for analyzing inverse local times

Abstract

In 1981 F. Knight published an article with a partial solution to a problem proposed by Ito-McKean see [Ito,[p.217]]. In this paper Knight, see [Knight], characterized the Levy measures of gap diffusions also known as quasi-diffusions. The proof is very elegant but it uses quite a lot functional analysis, more specifically spectral Krein Theory. We present a new proof of Knight's Theorem, defined at the beginning of the Introduction as well as the new proof of the same theorem referred as Theorem 2.6 in Section 2.