Stochastic fluctuations and the relaxation time in transient relativistic fluids

We argue that the ratio between the shear viscosity and the shear relaxation time, $\eta/\tau_\pi$, should be defined as a thermodynamic quantity obtained from the equal-time symmetric correlator of the shear-stress tensor. In kinetic theory, we show that this ratio does not depend on the type of interaction. Similarly, an exact expression for this ratio is obtained for holographic gauge theories. We also determine how stochastic fluctuations change $\eta/\tau_\pi$ in transient relativistic hydrodynamics and show that thermal fluctuations do not spoil causality and stability.