# Stochastic partial differential fluid equations as a diffusive limit of deterministic Lagrangian multi-time dynamics

## TL;DR

This paper demonstrates that stochastic fluid equations can be derived as a diffusive limit of deterministic multi-scale Lagrangian dynamics using homogenization theory, providing a rigorous justification for stochastic PDEs in fluid modeling.

## Contribution

It shows that stochastic fluid equations naturally emerge from a multi-scale decomposition of deterministic Lagrangian flows via homogenization, linking deterministic chaos to stochastic models.

## Key findings

- Derivation of stochastic fluid equations from deterministic multi-scale flows.
- Application of homogenization theory to justify stochastic PDEs.
- Identification of conditions under which fast small-scale dynamics lead to stochastic behavior.

## Abstract

In {\em{Holm}, Proc. Roy. Soc. A 471 (2015)} stochastic fluid equations were derived by employing a variational principle with an assumed stochastic Lagrangian particle dynamics. Here we show that the same stochastic Lagrangian dynamics naturally arises in a multi-scale decomposition of the deterministic Lagrangian flow map into a slow large-scale mean and a rapidly fluctuating small scale map. We employ homogenization theory to derive effective slow stochastic particle dynamics for the resolved mean part, thereby justifying stochastic fluid partial equations in the Eulerian formulation. To justify the application of rigorous homogenization theory, we assume mildly chaotic fast small-scale dynamics, as well as a centering condition. The latter requires that the mean of the fluctuating deviations is small, when pulled back to the mean flow.