Chaos, Ito-Stratonovich dilemma, and topological supersymmetry

ArXiv ID: 2512.21539 “View on arXiv”

Authors: Igor V. Ovchinnikov

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.

Keywords: stochastic differential equations, dynamical systems, topological field theory, chaos theory, supersymmetry

Complexity vs Empirical Score

• Math Complexity: 9.5/10
• Empirical Rigor: 1.0/10
• Quadrant: Lab Rats
• Why: The paper is deeply rooted in advanced mathematical physics, featuring extensive notation from topological field theory and cohomological supersymmetry, with no empirical data or backtesting presented.

  flowchart TD
    A["Research Goal<br/>Unify Stochastic Dynamics<br/>with Topological Field Theory"] --> B{"Key Methodology<br/>Generalized Transfer Operator &<br/>Supersymmetric Formalism"}
    B --> C["Computational Processes<br/>Apply STS Formalism to SDEs<br/>Analyze GTO Spectral Properties"]
    C --> D["Key Findings & Outcomes<br/>1. Topological Supersymmetry identified in all SDEs<br/>2. Stratonovich interpretation is the mathematically consistent choice<br/>3. Chaos (Positive Pressure) equals TS breakdown<br/>4. 1/f Noise explained via Goldstone theorem"]