Weak solutions for singular Lévy SDEs in the rough regime

Abstract

We introduce a new weak solution concept (called "rough weak solutions") for SDEs with distributional drift and additive stable Lévy noise and prove its equivalence to martingale solutions in the Young and rough regularity regime. In the rough regime this leads to a construction of certain rough stochastic sewing integrals involved. Moreover we prove that in the Young case, canonical weak solutions are well-posed and equivalent to rough weak solutions. However, in the rough case, canonical weak solutions are shown to be in general non-unique in law (which was our initial motivation for introducing rough weak solutions). We moreover prove a generalized Itô formula for rough weak solutions. This talk is based on joint work with Nicolas Perkowski.