  We develop a general formalism for introducing stochastic fluctuations around
thermodynamic equilibrium which takes into account, for the first time, recent
developments on the causality and stability properties of relativistic
hydrodynamic theories. The method is valid for any covariantly stable theory of
relativistic viscous fluid dynamics derived from a covariant maximum entropy
principle. We illustrate the formalism with some applications, showing how it
could be used to consistently introduce fluctuations in a model of relativistic
heat diffusion, and in conformally invariant Israel-Stewart theory in a general
hydrodynamic frame. The latter example is used to study the hydrodynamic frame
dependence of the symmetric two-point function of fluctuations of the
energy-momentum tensor.
