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

Xin Chen; Ana Bela Cruzeiro

doi:10.3934/proc.2013.2013.115

Article Contents

2013, Volume 2013: 115-121. Doi: 10.3934/proc.2013.2013.115 open access

This issue Previous Article New class of exact solutions for the equations of motion of a chain of $n$ rigid bodies Next Article On the uniqueness of singular solutions for a Hardy-Sobolev equation

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

Xin Chen^1, and
Ana Bela Cruzeiro^2,

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

Received: August 31, 2012

Revised: February 28, 2013

Published: October 2013

Abstract Related Papers Cited by

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:

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

Citation:

Related Papers

Cited by

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.