Fabian Andsem Harang

Abstract

The theory of regularization by noise is concerned with discovering the regularizing effects that stochastic or deterministic noise may have on ordinary and partial differential equations. In this presentation I will give a short review of recent results in this theory, as well as some applications.

In particular, I will begin to show a simple strategy to prove the regularizing effect of differential equations using the concept of averaging operators given as deterministic Riemann integral of a function along an irregular path. A specific example of such averaging operator can be obtained from  the local time of a stochastic processes. Using these operators we construct a solution theory for equations driven by a non-linear Young integral. I will then show some applications of these techniques towards the regularization of ill-posed and singular ODEs and PDEs perturbed by noise.