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Symmetries in the stochastic calculus of variations
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  • Published: March 1997

Symmetries in the stochastic calculus of variations

  • M. Thieullen1 &
  • J. C. Zambrini2 

Probability Theory and Related Fields volume 107, pages 401–427 (1997)Cite this article

  • 237 Accesses

  • 31 Citations

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Summary.

Given a stochastic action integral we define a notion of invariance of this action under a family of one parameter space-time transformations and a notion of prolonged transformations which extend the existing analogs in classical calculus of variations. We prove that a family of prolonged transformations leaves the action integral invariant if and only if it leaves invariant the heat equation associated to it as well as the structure of the extremals. We then prove a stochastic version of Noether theorem: to each family of transformations leaving the action invariant (or symmetries) we can associate a function which gives a martingale when taken along a process minimizing the action under endpoint constraints.

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Authors and Affiliations

  1. Laboratoire de Probabilités, Tour 56, 3ème étage, Université Paris 6, 4, Place Jussieu, F-75252 Paris Cedex 05, France, , , , , , FR

    M. Thieullen

  2. Grupo de Fisica-Matemática, Universidade de Lisboa, Av. Prof. Gama Pinto 2, 1699 Lisboa Codex, Portugal, , , , , , PT

    J. C. Zambrini

Authors
  1. M. Thieullen
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  2. J. C. Zambrini
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Received: 29 June 1996 / In revised form: 19 July 1996

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Thieullen, M., Zambrini, J. Symmetries in the stochastic calculus of variations. Probab Theory Relat Fields 107, 401–427 (1997). https://doi.org/10.1007/s004400050091

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  • Issue Date: March 1997

  • DOI: https://doi.org/10.1007/s004400050091

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  • Mathematics Subject Classification (1991): 60G44
  • 60J25
  • 60J60
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