  We consider a quantum Langevin kinetic equation for a system of fermions. We
first derive the Langevin force noise correlation functions in Landau's
Fermi-liquid kinetic theory from general considerations. We then use the
resulting equation to calculate the equilibrium dynamic structure factor in the
collisionless regime at low temperatures. The result is in agreement with the
conventional many-body result. We then use the theory to derive both the
fluctuating Navier-Stokes equations for a quantum fluid and the fluctuating
hydrodynamic equations for fermions in the presence of quenched disorder. We
also discuss the modifications needed for the fluctuating hydrodynamic
equations to describe an electron fluid with long-ranged interactions, and we
prove an H-theorem for the nonlinear Landau kinetic equation.
