Title: Stochastic partial differential fluid equations as a diffusive limit of deterministic Lagrangian multi-time dynamics,
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.