Noether's Theorem in the Stochastic Calculus of Variations

Abstract

We begin by presenting the classical deterministic problems of the calculus of variations, with emphasis on the necessary optimality conditions of Euler-Lagrange and the Noether theorem. As examples of application, we obtain the conservation laws of momentum and energy from mechanics, valid along the Euler-Lagrange extremals. We then introduce the stochastic calculus of variations, proving a recent stochastic Noether-type theorem obtained by Cresson. We end by pointing out an interesting open problem.