We introduce a family of stochastic models motivated by the study of
nonequilibrium steady states of fluid equations. These models decompose the
deterministic dynamics of interest into fundamental building blocks, i.e.,
minimal vector fields preserving some fundamental aspects of the original
dynamics. Randomness is injected by sequentially following each vector field
for a random amount of time. We show under general assumptions that these
random dynamics possess a unique invariant measure and converge almost surely
to the original, deterministic model in the small noise limit. We apply our
construction to the Lorenz-96 equations, often used in studies of chaos and
data assimilation, and Galerkin approximations of the 2D Euler and
Navier-Stokes equations. An interesting feature of the models developed is that
they apply directly to the conservative dynamics and not just those with
excitation and dissipation.
