  Causality is necessary for retarded Green's functions to remain retarded in
all inertial frames in relativity, which ensures that dissipation of
fluctuations is a Lorentz invariant concept. For first-order BDNK theories with
stochastic fluctuations, introduced via the Schwinger-Keldysh formalism, we
show that imposing causality and stability leads to correlation functions of
hydrodynamic fluctuations that only display the expected physical properties at
small frequencies and wavenumber, i.e., within the expected regime of validity
of the first-order approach. For second-order theories of Israel and Stewart
type, constructed using the information current such that entropy production is
always non-negative, a stochastic formulation is presented using the
Martin-Siggia-Rose approach where imposing causality and stability leads to
correlators with the desired properties. We also show how Green's functions can
be determined from such an action. We identify a $\mathbb{Z}_2$ symmetry,
analogous to the Kubo-Martin-Schwinger symmetry, under which this
Martin-Siggia-Rose action is invariant. This modified Kubo-Martin-Schwinger
symmetry provides a new guide for the effective action formulation of
hydrodynamic systems with dynamics not solely governed by conservation laws.
Furthermore, this symmetry ensures that the principle of detailed balance is
valid in a covariant manner. We employ the new symmetry to further clarify the
connection between the Schwinger-Keldysh and Martin-Siggia-Rose approaches,
establishing a precise link between these descriptions in second-order theories
of relativistic hydrodynamics. Finally, the modified Kubo-Martin-Schwinger
symmetry is used to determine the corresponding action describing diffusion in
Israel-Stewart theories in a general hydrodynamic frame.
