Minimal subharmonic functions and related integral representations

TL;DR

This paper establishes a Choquet-type integral representation for non-negative subharmonic functions of one-dimensional regular diffusions, enabling new insights into their structure and measure changes affecting long-term behavior.

Contribution

It introduces a novel integral representation for subharmonic functions and constructs Ito-Watanabe pairs, linking subharmonic functions with additive functionals and measure transformations.

Findings

Integral representation for subharmonic functions established

Construction of Ito-Watanabe pairs with local martingale property

Measure changes induce transience in the diffusion process

Abstract

A Choquet-type integral representation result for non-negative subharmonic functions of a one-dimensional regular diffusion is established. The representation allows in particular an integral equation for strictly positive subharmonic functions that is driven by the Revuz measure of the associated continuous additive functional. Moreover, via the aforementioned integral equation, one can construct an {\em \Ito-Watanabe pair} $(g,A)$ that consist of a subharmonic function $g$ and a continuous additive functional $A$ is with Revuz measure $\mu_A$ such that $g(X)\exp(-A)$ is a local martingale. Changes of measures associated with \Ito-Watanabe pairs are studied and shown to modify the long term behaviour of the original diffusion process to exhibit transience.