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Mass clumps in gravitational lens galaxies can perturb lensed images incharacteristic ways. Strong lens flux ratios have been used to constrain theamount of dark matter substructure in lens galaxies, and various otherobservables have been considered as additional probes of substructure. We studythe general theory of lensing with stochastic substructure in order tounderstand how lensing observables depend on the mass function and spatialdistribution of clumps. We find that magnification perturbations are mainlysensitive 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 ofthe clump mass function. Position perturbations are mainly sensitive to acharacteristic clump mass scale, namely m_eff = <m^2>/<m>, with some milddependence 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 delayperturbations are sensitive to the same characteristic mass, m_eff, and mainlydriven by the global population of clumps. While there is significant scatterin all lensing quantities, there are some non-trivial correlations that maycontain further information about the clump population. Our results indicatethat a joint analysis of multiple lens observables will offer qualitatively newconstraints on the mass function and spatial distribution of dark mattersubstructure in distant galaxies.

Gravitational lensing with stochastic substructure: Effects of the clump mass function and spatial distribution