The stochastic Hamilton-Jacobi equation

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September 2009, 1(3): 295-315. doi: 10.3934/jgm.2009.1.295

The stochastic Hamilton-Jacobi equation

Joan-Andreu Lázaro-Camí ^1, and Juan-Pablo Ortega ^2,

1.	Department of Mathematics, Imperial College, South Kensington Campus, London SW7 2AZ, United Kingdom
2.	Centre National de la Recherche Scientifique, Laboratoire de Mathématiques de Besançon, Université de Franche-Comté, UFR des Sciences et Techniques, 16, route de Gray, F-25030 Besançon cedex, France

Received May 2008 Revised August 2009 Published November 2009

Abstract
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We extend some aspects of the Hamilton-Jacobi theory to the category of stochastic Hamiltonian dynamical systems. More specifically, we show that the stochastic action satisfies the Hamilton-Jacobi equation when, as in the classical situation, it is written as a function of the configuration space using a regular Lagrangian submanifold. Additionally, we will use a variation of the Hamilton-Jacobi equation to characterize the generating functions of one-parameter groups of symplectomorphisms that allow to rewrite a given stochastic Hamiltonian system in a form whose solutions are very easy to find; this result recovers in the stochastic context the classical solution method by reduction to the equilibrium of a Hamiltonian system.

Keywords: Hamilton-Jacobi equation., Hamiltonian stochastic differential equation, stochastic differential equation.

Mathematics Subject Classification: Primary: 70H20, 81P2.

Citation: Joan-Andreu Lázaro-Camí, Juan-Pablo Ortega. The stochastic Hamilton-Jacobi equation. Journal of Geometric Mechanics, 2009, 1 (3) : 295-315. doi: 10.3934/jgm.2009.1.295

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