Prevalence of $\rho$-irregularity and related properties

We show that generic H\"older continuous functions are $\rho$-irregular. The property of $\rho$-irregularity has been first introduced by Catellier and Gubinelli (Stoc. Proc. Appl. 126, 2016) and plays a key role in the study of well-posedness for some classes of perturbed ODEs and PDEs. Genericity here is understood in the sense of prevalence. As a consequence we obtain several results on regularisation by noise "without probability", i.e. without committing to specific assumptions on the statistical properties of the perturbations. We also establish useful criteria for stochastic processes to be $\rho$-irregular and study in detail the geometric and analytic properties of $\rho$-irregular functions.