Quantitative Error Estimates for Learning Macroscopic Mobilities from Microscopic Fluctuations

TL;DR

This paper provides explicit quantitative error estimates linking microscopic fluctuations in particle systems to macroscopic mobility functions, enhancing understanding of the connection between microscopic and macroscopic stochastic models.

Contribution

It introduces new bounds for fluctuation-mobility discrepancies in particle systems and stochastic PDEs, including irregular coefficient cases like Dean-Kawasaki equations.

Findings

Explicit bounds for fluctuation-mobility discrepancy in particle systems

Error estimates for fluctuating hydrodynamic stochastic PDEs

Asymptotic behavior analysis for irregular coefficient SPDEs

Abstract

We develop quantitative error estimates connecting microscopic fluctuation of interacting particle systems with the mobilities of their hydrodynamic limits. Focusing on the Symmetric Simple Exclusion Process and systems of independent Brownian particles, we provide explicit bounds for the discrepancy between the quadratic variation of fluctuation fields and the corresponding mobilities, in terms of time and spatial discretization parameters. In addition, we establish analogous error estimates for a class of fluctuating hydrodynamic stochastic PDEs with regularized coefficients. For stochastic PDEs with irregular square-root type coefficients, including Dean-Kawasaki type equations, we further identify the asymptotic behavior of the associated fluctuation structures within the framework of renormalized kinetic solutions. Our results provide quantitative insights into the relationship between microscopic fluctuation mechanisms and macroscopic mobilities, and contribute to a structured comparison between discrete particle systems and continuum fluctuating hydrodynamic descriptions.