Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 21 (2025), 099, 25 pages      arXiv:2405.11051      https://doi.org/10.3842/SIGMA.2025.099

Darboux Transformation of Diffusion Processes

Alexey Kuznetsov and Minjian Yuan
Department of Mathematics and Statistics, York University, 4700 Keele Street, Toronto, ON, M3J 1P3, Canada

Received February 04, 2025, in final form November 12, 2025; Published online November 24, 2025

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.

Key words: diffusion process; Darboux transform; Sturm-Liouville theory; Markov semigroup; Doob's transform; Siegmund duality.

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