An analytic approach to regularization by noise phenomena for ODEs

One of the main questions in regularisation by noise literature is to understand whether an additive perturbation restores well-posedness of an ODE, i.e. under which conditions there exists a unique solution to $\dot{x}=b(x)+\dot{w}$ even if this is not the case for $w=0$. Davie first addressed the problem of identifying the analytical properties of a path $w$ which provide a regularising effect; Catellier and Gubinelli answered the problem by introducing the key concepts of averaging operators and nonlinear Young integrals. Remarkably, this allows to provide a consistent solution theory even when $b$ is merely distributional and to deduce that generic continuous functions have an arbitrarily high regularisation effect.

In this talk I will first review their work and then present its more recent extensions. Based on a joint work with Massimiliano Gubinelli.