Itô’s stochastic calculus and Heisenberg commutation relations |
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Authors: | Philippe Biane |
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Institution: | CNRS, Institut Gaspard Monge, Université Paris-Est, 77454 Marne-la-Vallée Cedex 2, France |
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Abstract: | Stochastic calculus and stochastic differential equations for Brownian motion were introduced by K. Itô in order to give a pathwise construction of diffusion processes. This calculus has deep connections with objects such as the Fock space and the Heisenberg canonical commutation relations, which have a central role in quantum physics. We review these connections, and give a brief introduction to the noncommutative extension of Itô’s stochastic integration due to Hudson and Parthasarathy. Then we apply this scheme to show how finite Markov chains can be constructed by solving stochastic differential equations, similar to diffusion equations, on the Fock space. |
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Keywords: | primary 60H05 60H10 secondary 46L53 |
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