Author(s): Marc Arnaudon
Institution: Institut de Mathématiques de Bordeaux (IMB), CNRS : UMR 5251, Université de Bordeaux, France
Slides: Arnaudon_Stochastic EulerPoincare reduction.pdf
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We will prove a Euler-Poincaré reduction theorem for stochastic processes taking values in a Lie group, which is a generalization of the Lagrangian version of reduction and its associated variational principles. We will also show examples of its application to the rigid body and to the group of diffeomorphisms, which includes the Navier-Stokes equation on a bounded domain and the Camassa-Holm equation.
References:M. Arnaudon, A.B. Cruzeiro and X. Chen, "Stochastic Euler-Poincaré Reduction", Journal of Mathematical Physics, to appear V. I. Arnold and B. Khesin, "Topological methods in hydrodynamics", Applied Math. Series 125, Springer (1998). J. M. Bismut, "Mécanique aléatoire", Lecture Notes in Mathematics, 866, Springer (1981). D.G. Ebin and J.E. Marsden, "Groups of diffeomorphisms and the motion of an incompressible fluid", Ann of Math. 92 (1970), 102--163. J. E. Marsden and T. S. Ratiu, "Introduction to Mechanics and Symmetry: a basic exposition of classical mechanical systems", Springer, Texts in Applied Math. (2003).
Marc Arnaudon was born in France in 1965. He graduated from Ecole Normale Supérieure de Paris, France, in 1991. He received the PhD degree in mathematics and the Habilitation à diriger des Recherches degree from Strasbourg University, France, in January 1994 and January 1998 respectively. After postdoctoral research and teaching at Strasbourg, he began in September 1999 a full professor position in the Department of Mathematics at Poitiers University, France, where he was the head of the Probability Research Group. In January 2013 he left Poitiers and joined the Department of Mathematics of Bordeaux University, France, where he is a full professor in mathematics.
Prof. Arnaudon is an expert in stochastic differential geometry and stochastic calculus in manifolds, he has published over 50 articles in mathematical and physical journals.