Reduction, reconstruction, and skew-product decomposition of symmetric stochastic differential equations

TL;DR

This paper develops methods for reducing and reconstructing solutions of symmetric stochastic differential equations, including local skew-product decompositions, with special focus on Hamiltonian systems and their conservation laws.

Contribution

It introduces reduction and reconstruction techniques for symmetric stochastic differential equations and adapts these methods to Hamiltonian systems considering conservation laws.

Findings

Methods for reduction and reconstruction of symmetric SDEs

Construction of local skew-product splittings near principal orbit types

Application to Hamiltonian systems with conservation laws

Abstract

We present reduction and reconstruction procedures for the solutions of symmetric stochastic differential equations, similar to those available for ordinary differential equations. Additionally, we use the local tangent-normal decomposition, available when the symmetry group is proper, to construct local skew-product splittings in a neighborhood of any point in the open and dense principal orbit type. The general methods introduced in the first part of the paper are then adapted to the Hamiltonian case, which is studied with special care and illustrated with several examples. The Hamiltonian category deserves a separate study since in that situation the presence of symmetries implies in most cases the existence of conservation laws, mathematically described via momentum maps, that should be taken into account in the analysis.