  Stochastic fluid dynamics governs the long time tails of hydrodynamic
correlation functions, and the critical slowing down of relaxation phenomena in
the vicinity of a critical point in the phase diagram. In this work we study
the role of multiplicative noise in stochastic fluid dynamics. Multiplicative
noise arises from the dependence of transport coefficients, such as the
diffusion constants for charge and momentum, on fluctuating hydrodynamic
variables. We study long time tails and relaxation in the diffusion of a
conserved density (model B), and a conserved density coupled to the transverse
momentum density (model H). Careful attention is paid to
fluctuation-dissipation relations. We observe that multiplicative noise
contributes at the same order as non-linear interactions in model B, but is a
higher order correction to the relaxation of a scalar density and the tail of
the stress tensor correlation function in model H.
