# Stochastic Euler-Poincaré reduction

## Abstract

We prove a Euler-Poincaré reduction theorem for stochastic processes taking values on a Lie group, which is a generalization of the reduction argument for the deterministic case [J. E. Marsden and T. S. Ratiu, Introduction to Mechanics and Symmetry: A Basic Exposition of Classical Mechanical Systems, Texts in Applied Mathematics (Springer, 2003)]. We also show examples of its application to SO(3) and to the group of diffeomorphisms, which includes the Navier-Stokes equation on a bounded domain and the Camassa-Holm equation.

- Authors:

- Institut de Mathématiques de Bordeaux (UMR 5251) Université Bordeaux 1 351, Cours de la Libération F33405 TALENCE Cedex (France)
- Grupo de Física-Matemática Univ. Lisboa, Av.Prof. Gama Pinto 2 1649-003 Lisboa (Portugal)
- GFMUL and Dep. de Matemática Instituto Superior Técnico (UL), Av. Rovisco Pais 1049-001 Lisboa (Portugal)

- Publication Date:

- OSTI Identifier:
- 22306025

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Journal of Mathematical Physics; Journal Volume: 55; Journal Issue: 8; Other Information: (c) 2014 AIP Publishing LLC; Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; CLASSICAL MECHANICS; LIE GROUPS; NAVIER-STOKES EQUATIONS; STOCHASTIC PROCESSES; SYMMETRY

### Citation Formats

```
Arnaudon, Marc, E-mail: marc.arnaudon@math.u-bordeaux1.fr, Chen, Xin, E-mail: chenxin-217@hotmail.com, and Cruzeiro, Ana Bela, E-mail: abcruz@math.ist.utl.pt.
```*Stochastic Euler-Poincaré reduction*. United States: N. p., 2014.
Web. doi:10.1063/1.4893357.

```
Arnaudon, Marc, E-mail: marc.arnaudon@math.u-bordeaux1.fr, Chen, Xin, E-mail: chenxin-217@hotmail.com, & Cruzeiro, Ana Bela, E-mail: abcruz@math.ist.utl.pt.
```*Stochastic Euler-Poincaré reduction*. United States. doi:10.1063/1.4893357.

```
Arnaudon, Marc, E-mail: marc.arnaudon@math.u-bordeaux1.fr, Chen, Xin, E-mail: chenxin-217@hotmail.com, and Cruzeiro, Ana Bela, E-mail: abcruz@math.ist.utl.pt. Fri .
"Stochastic Euler-Poincaré reduction". United States.
doi:10.1063/1.4893357.
```

```
@article{osti_22306025,
```

title = {Stochastic Euler-Poincaré reduction},

author = {Arnaudon, Marc, E-mail: marc.arnaudon@math.u-bordeaux1.fr and Chen, Xin, E-mail: chenxin-217@hotmail.com and Cruzeiro, Ana Bela, E-mail: abcruz@math.ist.utl.pt},

abstractNote = {We prove a Euler-Poincaré reduction theorem for stochastic processes taking values on a Lie group, which is a generalization of the reduction argument for the deterministic case [J. E. Marsden and T. S. Ratiu, Introduction to Mechanics and Symmetry: A Basic Exposition of Classical Mechanical Systems, Texts in Applied Mathematics (Springer, 2003)]. We also show examples of its application to SO(3) and to the group of diffeomorphisms, which includes the Navier-Stokes equation on a bounded domain and the Camassa-Holm equation.},

doi = {10.1063/1.4893357},

journal = {Journal of Mathematical Physics},

number = 8,

volume = 55,

place = {United States},

year = {Fri Aug 15 00:00:00 EDT 2014},

month = {Fri Aug 15 00:00:00 EDT 2014}

}

DOI: 10.1063/1.4893357

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