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 |
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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| _version_ | 1862715801123946496 |
|---|---|
| author | Kuznetsov, Alexey Yuan, Minjian |
| author_facet | Kuznetsov, Alexey Yuan, Minjian |
| citation_txt | Darboux Transformation of Diffusion Processes. Alexey Kuznetsov and Minjian Yuan. SIGMA 21 (2025), 099, 25 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | 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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| first_indexed | 2026-03-20T10:57:36Z |
| format | Article |
| fulltext | |
| id | nasplib_isofts_kiev_ua-123456789-214183 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-20T10:57:36Z |
| publishDate | 2025 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Kuznetsov, Alexey Yuan, Minjian 2026-02-20T07:54:19Z 2025 Darboux Transformation of Diffusion Processes. Alexey Kuznetsov and Minjian Yuan. SIGMA 21 (2025), 099, 25 pages 1815-0659 2020 Mathematics Subject Classification: 60J60; 60J35 arXiv:2405.11051 https://nasplib.isofts.kiev.ua/handle/123456789/214183 https://doi.org/10.3842/SIGMA.2025.099 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. The research was supported by the Natural Sciences and Engineering Research Council of Canada. The authors would like to thank Mateusz Kwaśnicki for stimulating discussions and for helping with the proof of Theorem 3.1. We are also grateful to anonymous referees for carefully reading the paper and for providing very helpful comments and suggestions. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Darboux Transformation of Diffusion Processes Article published earlier |
| spellingShingle | Darboux Transformation of Diffusion Processes Kuznetsov, Alexey Yuan, Minjian |
| title | Darboux Transformation of Diffusion Processes |
| title_full | Darboux Transformation of Diffusion Processes |
| title_fullStr | Darboux Transformation of Diffusion Processes |
| title_full_unstemmed | Darboux Transformation of Diffusion Processes |
| title_short | Darboux Transformation of Diffusion Processes |
| title_sort | darboux transformation of diffusion processes |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/214183 |
| work_keys_str_mv | AT kuznetsovalexey darbouxtransformationofdiffusionprocesses AT yuanminjian darbouxtransformationofdiffusionprocesses |