American Institute of Mathematical Sciences

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

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, 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.
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, ().   Google Scholar

[2]

M. Arnaudon and A.B. Cruzeiro, Lagrangian Navier Stokes diffusions on manifolds: variational principle and stability,, Bull. Sci. Math. 136 (2012), (2012), 857.   Google Scholar

[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.   Google Scholar

[4]

J.-M. Bismut, An introductory approach to duality in optimal stochastic control,, SIAM Rev, 20 (1978), 62.   Google Scholar

[5]

J.-M. Bismut, Mécanique Aléatoire,, Lecture Notes in Math. 866, (1981).   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[8]

A.B. Cruzeiro and Zh. Qian, Backward stochastic differential equations associated with the vorticity equations,, preprint, ().   Google Scholar

[9]

D. Ebin and J. Marsden, Groups of diffeomorphisms and the motion of an incompressible fluid,, Ann. of Math., 92 (1970), 102.   Google Scholar

[10]

A. Estrade and M. Pontier, Backward stochastic differential equations in a Lie group,, Séminaire de Probabilités, (2001), 241.   Google Scholar

[11]

W.H. Fleming and H.M. Soner, Controlled Markov processes and viscosity solutions,, Applications of Mathematics 25, (1993).   Google Scholar

[12]

B. Khesin, Groups and topology in the Euler hydrodynamics and KdV,, in Hamiltonian dynamical systems and applications, (2008), 93.   Google Scholar

[13]

J.A. Lázaro-Camí and J.P. Ortega, Stochastic Hamiltonian dynamical systems,, SIAM Review, 20 (1978), 65.   Google Scholar

[14]

J. Ma and J. Yong, Forward-backward stochastic differential equations and their applications,, Springer-Verlag, (2007).   Google Scholar

[15]

J.E. Marsden and T.S. Ratiu, Introduction to Mechanics and Symmetry,, Springer-Verlag, (2003).   Google Scholar

[16]

E. Pardoux and S. G. Peng, Adapted solution of a backward stochastic differential equation,, Systems Control Lett., 14 (1990), 55.   Google Scholar

[17]

J.-C. Zambrini, Variational processes and stochastic versions of mechanics,, Journal of Mathematical Physics, 27 (1986), 2307.   Google Scholar

[18]

J.-C. Zambrini, Stochastic Deformation of Classical Mechanics,, in these Proceedings., ().   Google Scholar

show all references

References:
[1]

M. Arnaudon, X. Chen and A.B. Cruzeiro, Stochastic Euler-Poincaré reduction,, preprint, ().   Google Scholar

[2]

M. Arnaudon and A.B. Cruzeiro, Lagrangian Navier Stokes diffusions on manifolds: variational principle and stability,, Bull. Sci. Math. 136 (2012), (2012), 857.   Google Scholar

[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.   Google Scholar

[4]

J.-M. Bismut, An introductory approach to duality in optimal stochastic control,, SIAM Rev, 20 (1978), 62.   Google Scholar

[5]

J.-M. Bismut, Mécanique Aléatoire,, Lecture Notes in Math. 866, (1981).   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[8]

A.B. Cruzeiro and Zh. Qian, Backward stochastic differential equations associated with the vorticity equations,, preprint, ().   Google Scholar

[9]

D. Ebin and J. Marsden, Groups of diffeomorphisms and the motion of an incompressible fluid,, Ann. of Math., 92 (1970), 102.   Google Scholar

[10]

A. Estrade and M. Pontier, Backward stochastic differential equations in a Lie group,, Séminaire de Probabilités, (2001), 241.   Google Scholar

[11]

W.H. Fleming and H.M. Soner, Controlled Markov processes and viscosity solutions,, Applications of Mathematics 25, (1993).   Google Scholar

[12]

B. Khesin, Groups and topology in the Euler hydrodynamics and KdV,, in Hamiltonian dynamical systems and applications, (2008), 93.   Google Scholar

[13]

J.A. Lázaro-Camí and J.P. Ortega, Stochastic Hamiltonian dynamical systems,, SIAM Review, 20 (1978), 65.   Google Scholar

[14]

J. Ma and J. Yong, Forward-backward stochastic differential equations and their applications,, Springer-Verlag, (2007).   Google Scholar

[15]

J.E. Marsden and T.S. Ratiu, Introduction to Mechanics and Symmetry,, Springer-Verlag, (2003).   Google Scholar

[16]

E. Pardoux and S. G. Peng, Adapted solution of a backward stochastic differential equation,, Systems Control Lett., 14 (1990), 55.   Google Scholar

[17]

J.-C. Zambrini, Variational processes and stochastic versions of mechanics,, Journal of Mathematical Physics, 27 (1986), 2307.   Google Scholar

[18]

J.-C. Zambrini, Stochastic Deformation of Classical Mechanics,, in these Proceedings., ().   Google Scholar

[1]

Yufeng Shi, Tianxiao Wang, Jiongmin Yong. Optimal control problems of forward-backward stochastic Volterra integral equations. Mathematical Control & Related Fields, 2015, 5 (3) : 613-649. doi: 10.3934/mcrf.2015.5.613
[2]

Jie Xiong, Shuaiqi Zhang, Yi Zhuang. A partially observed non-zero sum differential game of forward-backward stochastic differential equations and its application in finance. Mathematical Control & Related Fields, 2019, 9 (2) : 257-276. doi: 10.3934/mcrf.2019013
[3]

Dariusz Borkowski. Forward and backward filtering based on backward stochastic differential equations. Inverse Problems & Imaging, 2016, 10 (2) : 305-325. doi: 10.3934/ipi.2016002
[4]

Yuri Bakhtin. Lyapunov exponents for stochastic differential equations with infinite memory and application to stochastic Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2006, 6 (4) : 697-709. doi: 10.3934/dcdsb.2006.6.697
[5]

Ana Bela Cruzeiro. Navier-Stokes and stochastic Navier-Stokes equations via Lagrange multipliers. Journal of Geometric Mechanics, 2019, 11 (4) : 553-560. doi: 10.3934/jgm.2019027
[6]

Jasmina Djordjević, Svetlana Janković. Reflected backward stochastic differential equations with perturbations. Discrete & Continuous Dynamical Systems - A, 2018, 38 (4) : 1833-1848. doi: 10.3934/dcds.2018075
[7]

Jan A. Van Casteren. On backward stochastic differential equations in infinite dimensions. Discrete & Continuous Dynamical Systems - S, 2013, 6 (3) : 803-824. doi: 10.3934/dcdss.2013.6.803
[8]

Ying Hu, Shanjian Tang. Switching game of backward stochastic differential equations and associated system of obliquely reflected backward stochastic differential equations. Discrete & Continuous Dynamical Systems - A, 2015, 35 (11) : 5447-5465. doi: 10.3934/dcds.2015.35.5447
[9]

Anna Amirdjanova, Jie Xiong. Large deviation principle for a stochastic navier-Stokes equation in its vorticity form for a two-dimensional incompressible flow. Discrete & Continuous Dynamical Systems - B, 2006, 6 (4) : 651-666. doi: 10.3934/dcdsb.2006.6.651
[10]

Hakima Bessaih, Benedetta Ferrario. Statistical properties of stochastic 2D Navier-Stokes equations from linear models. Discrete & Continuous Dynamical Systems - B, 2016, 21 (9) : 2927-2947. doi: 10.3934/dcdsb.2016080
[11]

Lihuai Du, Ting Zhang. Local and global strong solution to the stochastic 3-D incompressible anisotropic Navier-Stokes equations. Discrete & Continuous Dynamical Systems - A, 2018, 38 (9) : 4745-4765. doi: 10.3934/dcds.2018209
[12]

Kumarasamy Sakthivel, Sivaguru S. Sritharan. Martingale solutions for stochastic Navier-Stokes equations driven by Lévy noise. Evolution Equations & Control Theory, 2012, 1 (2) : 355-392. doi: 10.3934/eect.2012.1.355
[13]

Takeshi Taniguchi. The existence and decay estimates of the solutions to $3$D stochastic Navier-Stokes equations with additive noise in an exterior domain. Discrete & Continuous Dynamical Systems - A, 2014, 34 (10) : 4323-4341. doi: 10.3934/dcds.2014.34.4323
[14]

G. Deugoué, T. Tachim Medjo. The Stochastic 3D globally modified Navier-Stokes equations: Existence, uniqueness and asymptotic behavior. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2593-2621. doi: 10.3934/cpaa.2018123
[15]

Xiao-Qian Jiang, Lun-Chuan Zhang. A pricing option approach based on backward stochastic differential equation theory. Discrete & Continuous Dynamical Systems - S, 2019, 12 (4&5) : 969-978. doi: 10.3934/dcdss.2019065
[16]

Qi Zhang, Huaizhong Zhao. Backward doubly stochastic differential equations with polynomial growth coefficients. Discrete & Continuous Dynamical Systems - A, 2015, 35 (11) : 5285-5315. doi: 10.3934/dcds.2015.35.5285
[17]

Yufeng Shi, Qingfeng Zhu. A Kneser-type theorem for backward doubly stochastic differential equations. Discrete & Continuous Dynamical Systems - B, 2010, 14 (4) : 1565-1579. doi: 10.3934/dcdsb.2010.14.1565
[18]

Yanqing Wang. A semidiscrete Galerkin scheme for backward stochastic parabolic differential equations. Mathematical Control & Related Fields, 2016, 6 (3) : 489-515. doi: 10.3934/mcrf.2016013
[19]

Weidong Zhao, Jinlei Wang, Shige Peng. Error estimates of the $\theta$-scheme for backward stochastic differential equations. Discrete & Continuous Dynamical Systems - B, 2009, 12 (4) : 905-924. doi: 10.3934/dcdsb.2009.12.905
[20]

Weidong Zhao, Yang Li, Guannan Zhang. A generalized $\theta$-scheme for solving backward stochastic differential equations. Discrete & Continuous Dynamical Systems - B, 2012, 17 (5) : 1585-1603. doi: 10.3934/dcdsb.2012.17.1585

 Impact Factor: 

Tools

Metrics

  • PDF downloads (20)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences