# 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, arXiv:1204.3922. [2] M. Arnaudon and A.B. Cruzeiro, Lagrangian Navier Stokes diffusions on manifolds: variational principle and stability, Bull. Sci. Math. 136 (2012), no. 8, 857-881. [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), 316-361. [4] J.-M. Bismut, An introductory approach to duality in optimal stochastic control, SIAM Rev, 20 (1978), 62-78. [5] J.-M. Bismut, Mécanique Aléatoire, Lecture Notes in Math. 866, Springer-Verlag (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), 255-269. [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), 4034-4060. [8] A.B. Cruzeiro and Zh. Qian, Backward stochastic differential equations associated with the vorticity equations, preprint, arXiv:1304.1319. [9] D. Ebin and J. Marsden, Groups of diffeomorphisms and the motion of an incompressible fluid, Ann. of Math., 92 (1970), 102-163. [10] A. Estrade and M. Pontier, Backward stochastic differential equations in a Lie group, Séminaire de Probabilités, XXXV, Lecture Notes in Math., 1755, Springer-Verlag (2001), 241-259. [11] W.H. Fleming and H.M. Soner, Controlled Markov processes and viscosity solutions, Applications of Mathematics 25, Springer-Verlag, New York (1993). [12] B. Khesin, Groups and topology in the Euler hydrodynamics and KdV, in Hamiltonian dynamical systems and applications, Ed. W.Craig, NATO Science series B, XVI Springer-Verlag (2008), 93-102. [13] J.A. Lázaro-Camí and J.P. Ortega, Stochastic Hamiltonian dynamical systems, SIAM Review, 20 (1978), 65-122. [14] J. Ma and J. Yong, Forward-backward stochastic differential equations and their applications, Springer-Verlag, Lecture Notes in Math. 1702, Springer-Verlag (2007). [15] J.E. Marsden and T.S. Ratiu, Introduction to Mechanics and Symmetry, Springer-Verlag, Texts in Applied Math. (2003). [16] E. Pardoux and S. G. Peng, Adapted solution of a backward stochastic differential equation, Systems Control Lett. , 14 (1990), 55-61. [17] J.-C. Zambrini, Variational processes and stochastic versions of mechanics, Journal of Mathematical Physics, 27 (1986), 2307-2330. [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, arXiv:1204.3922. [2] M. Arnaudon and A.B. Cruzeiro, Lagrangian Navier Stokes diffusions on manifolds: variational principle and stability, Bull. Sci. Math. 136 (2012), no. 8, 857-881. [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), 316-361. [4] J.-M. Bismut, An introductory approach to duality in optimal stochastic control, SIAM Rev, 20 (1978), 62-78. [5] J.-M. Bismut, Mécanique Aléatoire, Lecture Notes in Math. 866, Springer-Verlag (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), 255-269. [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), 4034-4060. [8] A.B. Cruzeiro and Zh. Qian, Backward stochastic differential equations associated with the vorticity equations, preprint, arXiv:1304.1319. [9] D. Ebin and J. Marsden, Groups of diffeomorphisms and the motion of an incompressible fluid, Ann. of Math., 92 (1970), 102-163. [10] A. Estrade and M. Pontier, Backward stochastic differential equations in a Lie group, Séminaire de Probabilités, XXXV, Lecture Notes in Math., 1755, Springer-Verlag (2001), 241-259. [11] W.H. Fleming and H.M. Soner, Controlled Markov processes and viscosity solutions, Applications of Mathematics 25, Springer-Verlag, New York (1993). [12] B. Khesin, Groups and topology in the Euler hydrodynamics and KdV, in Hamiltonian dynamical systems and applications, Ed. W.Craig, NATO Science series B, XVI Springer-Verlag (2008), 93-102. [13] J.A. Lázaro-Camí and J.P. Ortega, Stochastic Hamiltonian dynamical systems, SIAM Review, 20 (1978), 65-122. [14] J. Ma and J. Yong, Forward-backward stochastic differential equations and their applications, Springer-Verlag, Lecture Notes in Math. 1702, Springer-Verlag (2007). [15] J.E. Marsden and T.S. Ratiu, Introduction to Mechanics and Symmetry, Springer-Verlag, Texts in Applied Math. (2003). [16] E. Pardoux and S. G. Peng, Adapted solution of a backward stochastic differential equation, Systems Control Lett. , 14 (1990), 55-61. [17] J.-C. Zambrini, Variational processes and stochastic versions of mechanics, Journal of Mathematical Physics, 27 (1986), 2307-2330. [18] J.-C. Zambrini, Stochastic Deformation of Classical Mechanics, in these Proceedings.
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