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

Relativistic hydrodynamic fluctuations from an effective action: causality, stability, and the information current