C∞− regularization of ODEs perturbed by noise

Abstract

We study ordinary differential equations (ODEs) with vector fields given by general Schwartz distributions, and we show that if we perturb such an equation by adding an “infinitely regularizing” path, then it has a unique solution and it induces an infinitely smooth flow of diffeomorphisms. We also introduce a criterion under which the sample paths of a Gaussian process are infinitely regularizing, and we present two processes which satisfy our criterion. The results are based on the path-wise space–time regularity properties of local times, and solutions are constructed using the approach of Catellier–Gubinelli based on nonlinear Young integrals.