Chaos, Ito-Stratonovich dilemma, and topological supersymmetry

TL;DR

This paper explores the supersymmetric theory of stochastic dynamics (STS), revealing topological supersymmetry in stochastic differential equations, and links chaos, spectral properties, and interpretations of SDEs within a unified algebraic and topological framework.

Contribution

It introduces the supersymmetric theory of stochastic dynamics as a dual algebraic-topological framework for SDEs, highlighting the role of topological supersymmetry and its breakdown in chaos.

Findings

Positive pressure as a measure of chaos corresponds to supersymmetry breaking.

Stratonovich interpretation uniquely aligns with the GTOs of SDEs.

Potential explanation for 1/f noise via supersymmetry breaking.

Abstract

It was recently established that the formalism of the generalized transfer operator (GTO) of dynamical systems (DS) theory, applied to stochastic differential equations (SDEs) of arbitrary form, belongs to the family of cohomological topological field theories (TFT) -- a class of models at the intersection of algebraic topology and high-energy physics. This interdisciplinary approach, which can be called the supersymmetric theory of stochastic dynamics (STS), can be seen as an algebraic dual to the traditional set-theoretic framework of the DS theory, with its algebraic structure enabling the extension of some DS theory concepts to stochastic dynamics. Moreover, it reveals the presence of a topological supersymmetry (TS) in the GTOs of all SDEs. It also shows that among the various definitions of chaos, positive "pressure", defined as the logarithm of the GTO spectral radius, stands out as particularly meaningful from a physical perspective, as it corresponds to the spontaneous breakdown of TS on the TFT side. Via the Goldstone theorem, this definition has a potential to provide the long-sought explanation for the experimental signature of chaotic dynamics known as 1/f noise. Additionally, STS clarifies that among the various existing interpretations of SDEs, only the Stratonovich interpretation yields evolution operators that match the corresponding GTOs and, consequently, have a clear-cut mathematical meaning. Here, we discuss these and other aspects of STS from both the DS theory and TFT perspectives, focusing on links between these two fields and providing mathematical concepts with physical interpretations that may be useful in some contexts.