  We construct, for the first time, a Bemfica-Disconzi-Noronha-Kovtun (BDNK)
theory for linear stochastic fluctuations, which is proved to be mathematically
consistent, causal, and covariantly stable. The Martin-Siggia-Rose action is
shown to be bilocal in most cases, and the noise is not white. The presence of
nonhydrodynamic modes induces long-range correlations in the primary fluid
variables (temperature, chemical potential, and flow velocity). However,
correlators of conserved densities remain localized in space, and coincide with
those calculated within fluctuating Isreal-Stewart theory. We show that, in
some cases, there is a nonlocal change of variables that maps the
Israel-Stewart action into the BDNK action.
