Reduction and integrability of stochastic dynamical systems

2014/10/20 by Nguyen Tien Zung, Zung, Nguyen Tien, Thien, Nguyen Thanh
Mathematics · Physics and Astronomy · #37H10 #37J15 #37J35 #70H06 #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Homotopy and Cohomology in Algebraic Topology #Nonlinear Waves and Solitons

paper · doi:10.48550/arxiv.1410.5492

openalex publication_date 2014/10/20 · openalex created_date 2019/06/27 · openalex updated_date 2026/07/28

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.

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