Jacobian Bounds, Functional Inequalities, and Dispersion for Stochastic Differential Equations with Interaction

Abstract

We study evolutions of measures under flows of solutions to stochastic differential equations with interaction. Under suitable regularity assumptions on the coefficients, we prove moment bounds for the compact-set Jacobian norm. These bounds yield pathwise transfer of logarithmic Sobolev, concentration, and Talagrand transport inequalities from the initial measure to the random pushforward measure at time t. Finally, for a kernel model satisfying a pairwise dissipativity condition, we prove exponential decay of the expected dispersion around the random center of mass.