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.

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