# American Institute of Mathematical Sciences

2013, 2013(special): 115-121. doi: 10.3934/proc.2013.2013.115

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

 1 Grupo de Física-Matemática da Universidade de Lisboa, Av. Prof. Gama Pinto 2, 1649-003 Lisboa, Portugal 2 GFMUL and Departamento de Matemática IST-UTL, Av. Rovisco Pais, 1049-001 Lisboa, Portugal

Received  September 2012 Revised  March 2013 Published  November 2013

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.
Citation: Xin Chen, Ana Bela Cruzeiro. Stochastic geodesics and forward-backward stochastic differential equations on Lie groups. Conference Publications, 2013, 2013 (special) : 115-121. doi: 10.3934/proc.2013.2013.115
##### References:
 [1] M. Arnaudon, X. Chen and A.B. Cruzeiro, Stochastic Euler-Poincaré reduction,, preprint, (). [2] M. Arnaudon and A.B. Cruzeiro, Lagrangian Navier Stokes diffusions on manifolds: variational principle and stability,, Bull. Sci. Math. 136 (2012), (2012), 857. [3] V.I. Arnold, Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à l'hydrodynamique des fluides parfaits,, Ann. Inst. Fourier, 16 (1966), 316. [4] J.-M. Bismut, An introductory approach to duality in optimal stochastic control,, SIAM Rev, 20 (1978), 62. [5] J.-M. Bismut, Mécanique Aléatoire,, Lecture Notes in Math. 866, (1981). [6] F. Cipriano and A.B. Cruzeiro, Navier-Stokes equation and diffusions on the group of homeomorphisms of the torus,, Comm. Math. Phys., 275 (2007), 255. [7] A.B. Cruzeiro and E. Shamarova, Navier-Stokes equations and forward-backward SDEs on the group of diffeomorphisms of the torus,, Stoch. Proc. and their Applic., 119 (2009), 4034. [8] A.B. Cruzeiro and Zh. Qian, Backward stochastic differential equations associated with the vorticity equations,, preprint, (). [9] D. Ebin and J. Marsden, Groups of diffeomorphisms and the motion of an incompressible fluid,, Ann. of Math., 92 (1970), 102. [10] A. Estrade and M. Pontier, Backward stochastic differential equations in a Lie group,, Séminaire de Probabilités, (2001), 241. [11] W.H. Fleming and H.M. Soner, Controlled Markov processes and viscosity solutions,, Applications of Mathematics 25, (1993). [12] B. Khesin, Groups and topology in the Euler hydrodynamics and KdV,, in Hamiltonian dynamical systems and applications, (2008), 93. [13] J.A. Lázaro-Camí and J.P. Ortega, Stochastic Hamiltonian dynamical systems,, SIAM Review, 20 (1978), 65. [14] J. Ma and J. Yong, Forward-backward stochastic differential equations and their applications,, Springer-Verlag, (2007). [15] J.E. Marsden and T.S. Ratiu, Introduction to Mechanics and Symmetry,, Springer-Verlag, (2003). [16] E. Pardoux and S. G. Peng, Adapted solution of a backward stochastic differential equation,, Systems Control Lett., 14 (1990), 55. [17] J.-C. Zambrini, Variational processes and stochastic versions of mechanics,, Journal of Mathematical Physics, 27 (1986), 2307. [18] J.-C. Zambrini, Stochastic Deformation of Classical Mechanics,, in these Proceedings., ().

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##### References:
 [1] M. Arnaudon, X. Chen and A.B. Cruzeiro, Stochastic Euler-Poincaré reduction,, preprint, (). [2] M. Arnaudon and A.B. Cruzeiro, Lagrangian Navier Stokes diffusions on manifolds: variational principle and stability,, Bull. Sci. Math. 136 (2012), (2012), 857. [3] V.I. Arnold, Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à l'hydrodynamique des fluides parfaits,, Ann. Inst. Fourier, 16 (1966), 316. [4] J.-M. Bismut, An introductory approach to duality in optimal stochastic control,, SIAM Rev, 20 (1978), 62. [5] J.-M. Bismut, Mécanique Aléatoire,, Lecture Notes in Math. 866, (1981). [6] F. Cipriano and A.B. Cruzeiro, Navier-Stokes equation and diffusions on the group of homeomorphisms of the torus,, Comm. Math. Phys., 275 (2007), 255. [7] A.B. Cruzeiro and E. Shamarova, Navier-Stokes equations and forward-backward SDEs on the group of diffeomorphisms of the torus,, Stoch. Proc. and their Applic., 119 (2009), 4034. [8] A.B. Cruzeiro and Zh. Qian, Backward stochastic differential equations associated with the vorticity equations,, preprint, (). [9] D. Ebin and J. Marsden, Groups of diffeomorphisms and the motion of an incompressible fluid,, Ann. of Math., 92 (1970), 102. [10] A. Estrade and M. Pontier, Backward stochastic differential equations in a Lie group,, Séminaire de Probabilités, (2001), 241. [11] W.H. Fleming and H.M. Soner, Controlled Markov processes and viscosity solutions,, Applications of Mathematics 25, (1993). [12] B. Khesin, Groups and topology in the Euler hydrodynamics and KdV,, in Hamiltonian dynamical systems and applications, (2008), 93. [13] J.A. Lázaro-Camí and J.P. Ortega, Stochastic Hamiltonian dynamical systems,, SIAM Review, 20 (1978), 65. [14] J. Ma and J. Yong, Forward-backward stochastic differential equations and their applications,, Springer-Verlag, (2007). [15] J.E. Marsden and T.S. Ratiu, Introduction to Mechanics and Symmetry,, Springer-Verlag, (2003). [16] E. Pardoux and S. G. Peng, Adapted solution of a backward stochastic differential equation,, Systems Control Lett., 14 (1990), 55. [17] J.-C. Zambrini, Variational processes and stochastic versions of mechanics,, Journal of Mathematical Physics, 27 (1986), 2307. [18] J.-C. Zambrini, Stochastic Deformation of Classical Mechanics,, in these Proceedings., ().
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