On $ h $-transforms of one-dimensional diffusions stopped upon hitting zero

TL;DR

This paper studies three types of $h$-transforms of one-dimensional diffusions stopped at zero, exploring their properties and how they condition the process to avoid hitting zero, with implications for stochastic process theory.

Contribution

It introduces and analyzes new $h$-transforms based on ground state, scale function, and zero-resolvent for diffusions stopped at zero, expanding understanding of conditioned diffusions.

Findings

Characterization of $h$-transforms for diffusions avoiding zero

Properties of the $h$-transformed processes

Connections between different conditioning methods

Abstract

For a one-dimensional diffusion on an interval for which 0 is the regular-reflecting left boundary, three kinds of conditionings to avoid zero are studied. The limit processes are $ h $-transforms of the process stopped upon hitting zero, where $ h $'s are the ground state, the scale function, and the renormalized zero-resolvent. Several properties of the $ h $-transforms are investigated.