Stochastic hydrodynamics and long time tails of an expanding conformal charged fluid

We investigate the impact of hydrodynamic fluctuations on correlation functions in a scale invariant fluid with a conserved U(1) charge. The kinetic equations for the two-point functions of pressure, momentum, and heat energy densities are derived within the framework of stochastic hydrodynamics. The leading nonanalytic contributions to the energy-momentum tensor as well as the U(1) current are determined from the solutions to these kinetic equations. In the case of a static homogeneous background we show that the long time tails obtained from hydrokinetic equations reproduce the one-loop results derived from statistical field theory. We use these results to establish bounds on transport coefficients. We generalize the stochastic equation to a background flow undergoing Bjorken expansion. We compute the leading fractional power O((tau T)(-3/2)) correction to the U(1) current and compare with the first-order gradient term.