  In this paper we construct a new type of noise of fractional nature that has
a strong regularizing effect on differential equations. We consider an equation
with this noise with a highly irregular coefficient. We employ a new method to
prove existence and uniqueness of global strong solutions, where classical
methods fail because of the "roughness" and non-Markovianity of the driving
process. In addition, we prove the rather remarkable property that such
solutions are infinitely many times classically differentiable with respect to
the initial condition in spite of the vector field being discontinuous. The
technique used in this article corresponds to the Nash-Moser principle combined
with a new concept of "higher order averaging operators along highly fractal
stochastic curves". This approach may provide a general principle for the study
of regularization by noise effects in connection with important classes of
partial differential equations.
