  We study the transport properties of dilute electrolyte solutions on the
basis of the fluctuating hydrodynamic equation, which is a set of nonlinear
Langevin equations for the ion densities and flow velocity. The nonlinearity of
the Langevin equations generally leads to effective kinetic coefficients for
the deterministic dynamics of the average ion densities and flow velocity; the
effective coefficients generally differ from the counterparts in the Langevin
equations and are frequency-dependent. Using the path-integral formalism
involving auxiliary fields, we perform systematic perturbation calculations of
the effective kinetic coefficients for ion diffusion, shear viscosity, and
electrical conductivity, which govern the dynamics on the large length scales.
As novel contributions, we study the frequency dependence of the viscosity and
conductivity in the one-loop approximation. Regarding the conductivity at
finite frequencies, we derive the so-called electrophoretic part in addition to
the relaxation part, where the latter has been obtained by Debye and
Falkenhagen; it is predicted that the combination of these two parts gives rise
to the frequency $\omega_{\rm max}$ proportional to the salt density, at which
the real part of the conductivity exhibits a maximum. The zero-frequency limits
of the conductivity and shear viscosity coincide with the classical limiting
laws for dilute solutions, derived in different means by Debye, Falkenhagen,
and Onsager. As for the effective kinetic coefficients for slow ion diffusions
in large length scales, our straightforward calculation yields the cross
kinetic coefficient between cations and anions. Further, we discuss the
possibility of extending the present study to more concentrated solutions.
