  The recently developed effective field theory of fluctuations around thermal
equilibrium is used to compute late-time correlation functions of conserved
densities. Specializing to systems with a single conservation law, we find that
the diffusive pole is shifted in the presence of non-linear hydrodynamic
self-interactions, and that the density-density Green's function acquires a
branch point half way to the diffusive pole, at frequency $\omega=
-\frac{i}{2}Dk^2$. We discuss the relevance of diffusive fluctuations for
strongly correlated transport in condensed matter and cold atomic systems.
