The stochastic Hamilton-Jacobi equation

Joan-Andreu Lázaro-Camí; Juan-Pablo Ortega

doi:10.3934/jgm.2009.1.295

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2009, Volume 1, Issue 3: 295-315. Doi: 10.3934/jgm.2009.1.295

This issue Previous Article Cauchy problems for stationary Hamilton-Jacobi equations under mild regularity assumptions Next Article The Hamilton-Jacobi theory and the analogy between classical and quantum mechanics

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
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

Received: April 30, 2008

Revised: July 31, 2009

Published: August 2009

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Abstract

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:

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

Mathematics Subject Classification: Primary: 70H20, 81P20.

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