Article Contents
Article Contents

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

• 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.
Mathematics Subject Classification: Primary: 58J65, 70H30; Secondary: 60H10, 49Q20.

 Citation:

•  [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.
Open Access Under a Creative Commons license