A theory of generalised coordinates for stochastic differential equations (2409.15532v2)

Abstract: Stochastic differential equations are ubiquitous modelling tools in physics and the sciences. In most modelling scenarios, random fluctuations driving dynamics or motion have some non-trivial temporal correlation structure, which renders the SDE non-Markovian; a phenomenon commonly known as `colored'' noise. Thus, an important objective is to develop effective tools for mathematically and numerically studying (possibly non-Markovian) SDEs. In this report, we formalise a mathematical theory for analysing and numerically studying SDEs based on so-calledgeneralised coordinates of motion'. Like the theory of rough paths, we analyse SDEs pathwise for any given realisation of the noise, not solely probabilistically. Like the established theory of Markovian realisation, we realise non-Markovian SDEs as a Markov process in an extended space. Unlike the established theory of Markovian realisation however, the Markovian realisations here are accurate on short timescales and may be exact globally in time, when flows and fluctuations are analytic. This theory is exact for SDEs with analytic flows and fluctuations, and is approximate when flows and fluctuations are differentiable. It provides useful analysis tools, which we employ to solve linear SDEs with analytic fluctuations. It may also be useful for studying rougher SDEs, as these may be identified as the limit of smoother ones. This theory supplies effective, computationally straightforward methods for simulation, filtering and control of SDEs; amongst others, we re-derive generalised Bayesian filtering, a state-of-the-art method for time-series analysis. Looking forward, this report suggests that generalised coordinates have far-reaching applications throughout stochastic differential equations.

• The paper introduces generalized coordinates that reformulate SDEs into higher-dimensional systems incorporating derivatives beyond position.
• It demonstrates enhanced pathwise analysis and efficient numerical methods like zigzag integration for accurate short-term dynamic predictions.
• The framework improves Bayesian filtering applications and paves the way for advanced stochastic control and multi-scale system research.

A Theory of Generalized Coordinates for Stochastic Differential Equations

The research paper entitled "A Theory of Generalized Coordinates for Stochastic Differential Equations" by Da Costa et al. presents a novel mathematical framework designed to analyze and numerically solve Stochastic Differential Equations (SDEs). These equations are fundamental in modeling physical systems characterized by random dynamics, especially when these dynamics exhibit non-Markovian behavior due to temporal correlations in the noise driving the system.

Core Concepts and Methodology

The key innovation of this paper is the introduction of generalized coordinates of motion for the pathwise analysis of SDEs. This technique involves reformulating an SDE as a higher-dimensional dynamical system that encompasses not only the position but also higher derivatives such as velocity, acceleration, and so on—up to an arbitrary order. The authors leverage concepts akin to Taylor expansions, constructing solutions that transform the paper of systems subjected to noise into mathematically tractable problems under both deterministic and stochastic perspectives.

Pathwise Analysis and Markovian Realization: The paper draws parallels to the theory of rough paths by enabling pathwise analysis, meaning that SDEs are treated with respect to individual realizations of noise rather than relying solely on statistical properties. It extends the Markovian realization technique by embedding non-Markovian systems into an extended Markov process within this space of generalized coordinates, claiming accuracy on short timescales and potential global exactness when flows are analytic.

Numerical Results and Techniques

The paper provides substantial numerical results demonstrating the practical implications of their theoretical findings. Key results include:

• Linear SDEs with Analytic Noise: The paper offers a comprehensive analysis of linear SDEs driven by analytic fluctuations, including explicit mean and covariance characterizations under the generalized coordinate framework.
• Numerical Stability and Accuracy: The numerical methods proposed, particularly the zigzag integration methods and variants, are showcased for their ability to accurately capture the dynamics of complex systems over short intervals.
• Bayesian Filtering Applications: Through the generalization of Bayesian filtering to this new framework, the research presents improved methods for state estimation in dynamic systems—an essential tool in fields ranging from neuroimaging to robotics.

Implications and Future Directions

The paper speculates on significant practical and theoretical implications, suggesting that generalized coordinates offer versatile and computationally efficient methods for simulating and controlling SDE-driven systems. Future directions could include exploring multi-scale systems, further integrating rough path theory, and elaborating stochastic control uses through active inference principles.

The framework's adaptability to non-additive noise scenarios or time-varying non-stationary processes also presents intriguing prospects for research. The integration of more advanced machine learning paradigms or AI-focused applications could leverage these findings, particularly in environments where the noise characteristics and system dynamics are highly complex and data-intensive.

In conclusion, Da Costa et al. introduce a mathematically robust and computationally innovative approach to dealing with the intrinsic randomness of differential systems, promising improvements in various scientific domains reliant on SDE modeling for prediction and analysis.