Abstract: External perturbations applied to an ill-posed differential equation with highly irregular vector fields can restore well-posed problem, commonly referred to as the regularization by noise phenomenon. In 2021, Lucio Galeati and Massimiliano Gubinelli developed a "noiseless" regularization by noise framework which, unlike classical stochastic approaches relying on Itô calculus, does not depend on any underlying probability law, instead working path-by-path.

 

We first show how sufficiently irregular perturbations (most notably fractional Brownian motion of low Hurst parameter) can regularize irregular vector fields. With this, we then establish existence and uniqueness of solutions of the perturbed ODEs, with additional regularity on the flow depending on the roughness of the perturbation. If time permits, we discuss applications to transport-type equations.