“Regularisation by noise and notions of irregularity”
Venerdì 26 Giugno 2020, ore 11:00 - Zoom - Lucio Galeati (University of Bonn)
A natural question in the field of regularisation by noise phenomena is to understand under which conditions on a path $w$, the additively perturbed ODE $x’=b(x)+w’$ is well-posed, even when its unperturbed counterpart $x’=b(x)$ is not. In order to answer this question, Catellier and Gubinelli (SPA 2016) introduced the theory of nonlinear Young integrals and the concept of $\rho$-irregularity of $w$; they also proved that fBm paths are $\rho$-irregular. In this talk, I will first review their work and then present its more recent extensions.
The property of $\rho$-irregularity can be linked to the regularity of the local time of $w$ and, in the case of Gaussian processes, can be established under a suitable local non-determinism (LND) condition. Moreover, it can be shown that $\rho$-irregular paths are necessarily nowhere $\alpha$-Holder continuous, for suitable values $\alpha=\alpha(\rho)$. This gives a rigorous proof of the frequently observed paradigm “the rougher the noise, the better the regularisation”.
Based on joint works with Massiliano Gubinelli.