Lemme de coherence et théorème de Noether stochastique

Jacky Cresson (1) , Sébastien Darses (1)

1 LMB - Laboratoire de Mathématiques de Besançon (UMR 6623)

The stochastic embedding procedure associates a stochastic Euler-Lagrange equation (SEL) to the standard Euler-Lagrange equation (EL). Can we derive (SEL) from a generalized least action principle? To address this question, we develop a stochastic calculus of variation initiated by Yasue. We give a stochastic analog F of the lagrangian action functional. We introduce a notion of stationarity according to which the solutions of (SEL) are the stationary points of F. This notion of stationarity brings coherence to stochastic calculus of variation with respect to stochastic embedding. Finally, we prove a stochastic Noether theorem which introduces an original notion of stochastic first integral.

Mots clés: Stochastic calculus of variation; Stochastic Noether theorem; stochastic first integral; stochastic embedding

Domaines: Probabilités [math.PR]; Systèmes dynamiques [math.DS]

Jacky Cresson, Sébastien Darses. Lemme de coherence et théorème de Noether stochastique. 2005. ⟨hal-00012929⟩