Noether's Theorem in the Stochastic Calculus of Variations

TL;DR

This paper explores the classical calculus of variations and extends Noether's theorem into the stochastic calculus of variations, highlighting conservation laws and presenting recent theoretical advancements.

Contribution

It introduces a stochastic version of Noether's theorem, connecting symmetry principles to conservation laws in stochastic systems, building on recent research.

Findings

Derivation of conservation laws in classical mechanics

Introduction of stochastic Noether's theorem by Cresson

Identification of open problems in 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.