Continuitiy properties of the extension of a locally Lipschitz continuous map to the space of probability measures

Authors: Heinz W. Engl and Anton Wakolbinger

Abstract: For a locally Lipschitz continuous mapping F between metric spaces Z and X and probability measures mu,nu on Z, bounds for the (Prokhorov and boundes Lipschitz) distance of mu F^-1 and nu F^-1 are obtained in terms of the distance of mu and nu, the growth of the local Lipschitz constants of F, and a tail estimate of mu. As applications, we estimate convergence rates of approximate solutions of stochastic differential equations and obtain conditions on the speed of convergence of regularization parameters which guarantee convergence in distribution for the solutions of a random integral equation of the first kind.