The Wronskian parameterizes the class of diffusions with a given distribution at a random time

TL;DR

This paper characterizes all one-dimensional diffusions that match a specified distribution at a random exponential time, extending classical diffusion theory results and illustrating with Brownian motion.

Contribution

It provides a complete characterization of diffusions with a given distribution at an exponential time, using the Wronskian parameterization.

Findings

Classifies diffusions with a specified distribution at exponential times

Uses classical diffusion theory for characterization

Includes Brownian motion as a special case

Abstract

We provide a complete characterization of the class of one-dimensional time-homogeneous diffusions consistent with a given law at an exponentially distributed time using classical results in diffusion theory. To illustrate we characterize the class of diffusions with the same distribution as Brownian motion at an exponentially distributed time.