Stochastic geodesics and forward-backward stochastic differential equations on Lie groups

Pages: 115 - 121, Issue special, November 2013

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Xin Chen - Grupo de Física-Matemática da Universidade de Lisboa, Av. Prof. Gama Pinto 2, 1649-003 Lisboa, Portugal (email)
Ana Bela Cruzeiro - GFMUL and Departamento de Matemática IST-UTL, Av. Rovisco Pais, 1049-001 Lisboa, Portugal (email)

Abstract: We describe how to generalize to the stochastic case the notion of geodesic on a Lie group equipped with an invariant metric. As second order equations (in time), stochastic geodesics are characterized in terms of stochastic forward-backward differential systems.
    When the group is the diffeomorphisms group this corresponds to a probabilistic description of the Navier-Stokes equations.

Keywords:  Forward-backward stochastic differential equations, stochastic variational principles, stochastic geodesics on Lie groups, Navier-Stokes equation, Lagrangian stochastic flows.
Mathematics Subject Classification:  Primary: 58J65, 70H30; Secondary: 60H10, 49Q20.

Received: September 2012;      Revised: March 2013;      Published: November 2013.