Darboux Transformation of Diffusion Processes

TL;DR

This paper explores Darboux transformations applied to diffusion processes, linking differential operator techniques with Markov semigroups, and provides explicit formulas and examples involving Brownian motion and Ornstein-Uhlenbeck processes.

Contribution

It introduces a novel approach to Darboux transformation for diffusion processes using Doob's $h$-transform and duality, with explicit formulas connecting transition densities.

Findings

Derived a simple formula relating transition densities of original and transformed processes

Explicitly computed transition densities for Brownian motion and Ornstein-Uhlenbeck processes

Provided spectral representations for the transformed processes

Abstract

Darboux transformation of a second-order linear differential operator is a well-known technique with many applications in mathematics and physics. We study Darboux transformation from the point of view of Markov semigroups of diffusion processes. We construct the Darboux transform of a diffusion process through a combination of Doob's $h$-transform and a version of Siegmund duality. Our main result is a simple formula that connects transition probability densities of the two processes. We provide several examples of Darboux transformed diffusion processes related to Brownian motion and Ornstein-Uhlenbeck process. For these examples, we compute explicitly the transition probability density and derive its spectral representation.