C-infinity regularization of ODEs perturbed by noise (2003.05816v2)

Abstract: We study 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 non-linear Young integrals.