Minimal subharmonic functions and related integral representations

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.