The stochastic Hamilton-Jacobi equation

TL;DR

This paper extends Hamilton-Jacobi theory to stochastic Hamiltonian systems, showing that the stochastic action satisfies a Hamilton-Jacobi equation and using it to simplify the integration of such systems.

Contribution

It introduces a stochastic Hamilton-Jacobi equation and a method to find solutions by reducing systems to equilibrium, adapting classical techniques to stochastic contexts.

Findings

Stochastic action satisfies a Hamilton-Jacobi equation.

Characterization of generating functions for symplectomorphisms in stochastic systems.

Reduction of stochastic Hamiltonian systems to equilibrium for easier solutions.

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.