Momentum Maps and Stochastic Clebsch Action Principles

TL;DR

This paper develops a stochastic calculus framework for Lie algebra actions on manifolds, introducing a stochastic Clebsch principle that simplifies stochastic Hamiltonian equations via momentum maps.

Contribution

It introduces a stochastic Clebsch action principle coupling noise through momentum maps, leading to simplified stochastic Hamilton equations and new insights into their structure.

Findings

Derived stochastic differential equations for Lie algebra actions.

Established the stochastic Clebsch variational principle with momentum map coupling.

Connected Itô correction to a double Poisson bracket.

Abstract

We derive stochastic differential equations whose solutions follow the flow of a stochastic nonlinear Lie algebra operation on a configuration manifold. For this purpose, we develop a stochastic Clebsch action principle, in which the noise couples to the phase space variables through a momentum map. This special coupling simplifies the structure of the resulting stochastic Hamilton equations for the momentum map. In particular, these stochastic Hamilton equations collectivize for Hamiltonians that depend only on the momentum map variable. The Stratonovich equations are derived from the Clebsch variational principle and then converted into It\^o form. In comparing the Stratonovich and It\^o forms of the stochastic dynamical equations governing the components of the momentum map, we find that the It\^o contraction term turns out to be a double Poisson bracket. Finally, we present the stochastic Hamiltonian formulation of the collectivized momentum map dynamics and derive the corresponding Kolmogorov forward and backward equations.