Interpretation of generalized Langevin equations

TL;DR

This paper extends the interpretive framework of stochastic differential equations to include nonlinear dependencies on random variables, enabling better modeling of complex systems like particles in turbulence.

Contribution

It introduces a method to interpret nonlinear stochastic systems as equivalent Itô processes, broadening the scope of analyzable models beyond traditional linear assumptions.

Findings

Successfully applied the method to a toy system.

Demonstrated the approach on a physical system involving particle velocity in turbulence.

Enables analysis of previously intractable nonlinear stochastic models.

Abstract

Many real-world systems exhibit ``noisy'' evolution in time; interpreting their finitely-sampled behavior as arising from continuous-time processes (in the It\^o or Stratonovich sense) has led to significant success in modeling and analysis in a wide variety of fields. Yet such interpretation hinges on a fundamental linear separation of randomness from determinism in the underlying dynamics. Here we propose some theoretical systems which resist easy and self-consistent interpretation into this well-defined class of equations, requiring an expansion of the interpretive framework. We argue that a wider class of stochastic differential equations, where evolution depends nonlinearly on a random or effectively-random quantity, may be consistently interpreted and in fact exhibit finite-time stochastic behavior in line with an equivalent It\^o process, at which point many existing numerical and analytical techniques may be used. We put forward a method for this conversion, and demonstrate its use on both a toy system and on a system of direct physical relevance: the velocity of a meso-scale particle suspended in a turbulent fluid. This work enables the theoretical and numerical examination of a wide class of mathematical models which might otherwise be oversimplified due to a lack of appropriate tools.