Noether's Theorem in the Stochastic Calculus of Variations

Adilson C. M. Barros, Delfim F. M. Torres

Publication date: 27 August 2012

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.

Mathematics Subject Classification ID

General theory of stochastic processes (60G07) Generation, random and stochastic difference and differential equations (37H10) Variational principles of physics (49S05) Optimality conditions for free problems in one independent variable (49K05)

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