Reduction and integrability of stochastic dynamical systems

TL;DR

This paper explores the geometric properties of stochastic dynamical systems, focusing on symmetries, reduction, and integrability, and extends classical results to stochastic contexts.

Contribution

It introduces conditions for reduction and projectability of SDSs, and extends integrability and Liouville torus actions to stochastic systems.

Findings

SDSs with diffusion-wise symmetry can be reduced to quotient spaces.

Necessary and sufficient conditions for SDS projectability are established.

Connections between integrable SDSs and compatible Riemannian metrics are demonstrated.

Abstract

This paper is devoted to the study of qualitative geometrical properties of stochastic dynamical systems, namely their symmetries, reduction and integrability. In particular, we show that an SDS which is diffusion-wise symmetric with respect to a proper Lie group action can be diffusion-wise reduced to an SDS on the quotient space. We also show necessary and sufficient conditions for an SDS to be projectable via a surjective map. We then introduce the notion of integrability of SDS's, and extend the results about the existence and structure-preserving property of Liouville torus actions from the classical case to the case of integrable SDS's. We also show how integrable SDS's are related to compatible families of integrable Riemannian metrics on manifolds.