  We study spectral properties of the Fokker-Planck operator that represents
particles moving via a combination of diffusion and advection in a
time-independent random velocity field, presenting in detail work outlined
elsewhere [J. T. Chalker and Z. J. Wang, Phys. Rev. Lett. {\bf 79}, 1797
(1997)]. We calculate analytically the ensemble-averaged one-particle Green
function and the eigenvalue density for this Fokker-Planck operator, using a
diagrammatic expansion developed for resolvents of non-Hermitian random
operators, together with a mean-field approximation (the self-consistent Born
approximation) which is well-controlled in the weak-disorder regime for
dimension d>2. The eigenvalue density in the complex plane is non-zero within a
wedge that encloses the negative real axis. Particle motion is diffusive at
long times, but for short times we find a novel time-dependence of the
mean-square displacement, $<r^2> \sim t^{2/d}$ in dimension d>2, associated
with the imaginary parts of eigenvalues.
