Prevalence of $\rho$-irregularity and related properties

TL;DR

This paper proves that most H"older continuous functions are $ ho$-irregular, which is crucial for understanding well-posedness in perturbed differential equations, and develops criteria for stochastic processes to exhibit this property.

Contribution

It demonstrates the prevalence of $ ho$-irregularity among H"older functions and provides criteria for stochastic processes to be $ ho$-irregular, advancing the theory without relying on probabilistic assumptions.

Findings

Most H"older continuous functions are $ ho$-irregular.

Criteria established for stochastic processes to be $ ho$-irregular.

Analysis of geometric and analytic properties of $ ho$-irregular functions.

Abstract

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.