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

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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

Abstract
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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: Lagrangian stochastic flows., Forward-backward stochastic differential equations, Navier-Stokes equation, stochastic variational principles, stochastic geodesics on Lie groups.

Mathematics Subject Classification: Primary: 58J65, 70H30; Secondary: 60H10, 49Q2.

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., ().

show all references

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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