Abstract
We investigate the symmetries of a class of diffusions processes (“Bernstein’s reciprocal” processes) introduced in the eighties for the solution of a problem stated by Schrödinger in 1931. Those diffusions satisfy two unusual properties. Although typically not time-homogeneous, they are time reversible. Also their infinitesimal coefficients are specific functions of positive solutions of time adjoint parabolic equations. The symmetries of these PDEs will therefore be transformed into symmetries of the diffusions and provide relations between them hard to guess otherwise. We shall use an algebraico-geometric method (“of isovectors”) and mention applications in finance and mathematical physics. As can be expected Schrödinger’s initial motivation was quantum mechanics.
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Acknowledgements
Both authors are extremely grateful to Professors Marco Fuhrman, Elisa Mastrogiacomo, Paola Morando and Stefania Ugolini for their excellent organization of the conference and their wonderful hospitality. Professor Jean-Claude Zambrini was kind enough to read a preliminary version of our manuscript; his comments and suggestions led to numerous improvements in the structure and style of our text. We thank him heartily for his efforts.
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Lescot, P., Valade, L. (2021). Bernstein Processes, Isovectors and Mechanics. In: Ugolini, S., Fuhrman, M., Mastrogiacomo, E., Morando, P., Rüdiger, B. (eds) Geometry and Invariance in Stochastic Dynamics. RTISD19 2019. Springer Proceedings in Mathematics & Statistics, vol 378. Springer, Cham. https://doi.org/10.1007/978-3-030-87432-2_11
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DOI: https://doi.org/10.1007/978-3-030-87432-2_11
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