Darboux Transformation of Diffusion Processes
Darboux transformation of a second-order linear differential operator is a well-known technique with many applications in mathematics and physics. We study the Darboux transformation from the point of view of Markov semigroups of diffusion processes. We construct the Darboux transform of a diffusion...
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| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Date: | 2025 |
| Main Authors: | , |
| Format: | Article |
| Language: | English |
| Published: |
Інститут математики НАН України
2025
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/214183 |
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| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | Darboux Transformation of Diffusion Processes. Alexey Kuznetsov and Minjian Yuan. SIGMA 21 (2025), 099, 25 pages |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| Summary: | Darboux transformation of a second-order linear differential operator is a well-known technique with many applications in mathematics and physics. We study the 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 the Ornstein-Uhlenbeck process. For these examples, we compute the transition probability density explicitly and derive its spectral representation.
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| ISSN: | 1815-0659 |