Эта публикация цитируется в 1 статье

Darboux Transformation of Diffusion Processes

Alexey Kuznetsov, Minjian Yuan

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

Аннотация: 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.

Ключевые слова: diffusion process, Darboux transform, Sturm–Liouville theory, Markov semigroup, Doob's transform, Siegmund duality.

MSC: 60J60, 60J35

Поступила: 4 февраля 2025 г.; в окончательном варианте 12 ноября 2025 г.; опубликована 24 ноября 2025 г.

Язык публикации: английский

DOI: 10.3842/SIGMA.2025.099

ArXiv: 2405.11051