# American Institute of Mathematical Sciences

June  2016, 8(2): 221-233. doi: 10.3934/jgm.2016005

## A weak approach to the stochastic deformation of classical mechanics

 1 Université Paris-Dauphine, Place du Maréchal De Lattre De Tassigny, 75775 Paris Cedex 16, France 2 GFM, Group of Mathematical Physics University of Lisbon, Department of Mathematics Faculty of Sciences, Campo Grande, Edifcio C6 PT-1749-016 Lisboa, Portugal

Received  December 2014 Revised  January 2016 Published  June 2016

We establish a transfer principle, providing a canonical form of dynamics to stochastic models, inherited from their classical counterparts. The stochastic deformation of Euler$-$Lagrange conditions, and the associated Hamiltonian formulations, are given as conditions on laws of processes. This framework is shown to encompass classical models, and the so-called Schrödinger bridges. Other applications and perspectives are provided.
Citation: Rémi Lassalle, Jean Claude Zambrini. A weak approach to the stochastic deformation of classical mechanics. Journal of Geometric Mechanics, 2016, 8 (2) : 221-233. doi: 10.3934/jgm.2016005
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