Darboux Transformation of Diffusion Processes
Probability
2025-11-26 v2
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 -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.
Cite
@article{arxiv.2405.11051,
title = {Darboux Transformation of Diffusion Processes},
author = {Alexey Kuznetsov and Minjian Yuan},
journal= {arXiv preprint arXiv:2405.11051},
year = {2025}
}