  Mass clumps in gravitational lens galaxies can perturb lensed images in
characteristic ways. Strong lens flux ratios have been used to constrain the
amount of dark matter substructure in lens galaxies, and various other
observables have been considered as additional probes of substructure. We study
the general theory of lensing with stochastic substructure in order to
understand how lensing observables depend on the mass function and spatial
distribution of clumps. We find that magnification perturbations are mainly
sensitive to the total mass in substructure projected near the lensed images;
when the source is small, flux ratios are not very sensitive to the shape of
the clump mass function. Position perturbations are mainly sensitive to a
characteristic clump mass scale, namely m_eff = <m^2>/<m>, with some mild
dependence on other mass moments when the spatial distribution is not uniform.
They have contributions from both "local" and "global" populations of clumps
(i.e., those projected near the images, and those farther away). Time delay
perturbations are sensitive to the same characteristic mass, m_eff, and mainly
driven by the global population of clumps. While there is significant scatter
in all lensing quantities, there are some non-trivial correlations that may
contain further information about the clump population. Our results indicate
that a joint analysis of multiple lens observables will offer qualitatively new
constraints on the mass function and spatial distribution of dark matter
substructure in distant galaxies.
