  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.
