Rough weak solutions for singular L\'evy SDEs

TL;DR

This paper introduces a new concept of rough weak solutions for singular Lévy SDEs, establishing their equivalence to existing martingale solutions in certain regimes and analyzing well-posedness and uniqueness issues.

Contribution

It develops the theory of rough weak solutions for singular Lévy SDEs, including construction, equivalence, and analysis of well-posedness and non-uniqueness.

Findings

Rough weak solutions are equivalent to martingale solutions in the Young and rough regimes.

Canonical weak solutions are well-posed in the Young regime.

Counterexample demonstrates non-uniqueness in the rough regime.

Abstract

We introduce a weak solution concept (called "rough weak solutions") for singular SDEs with additive alpha-stable L\'evy noise (including the Brownian noise case) and prove its equivalence to martingale solutions from Kremp, Perkowski '22 in the Young and rough regularity regime. In the rough regime this leads to the construction of certain rough stochastic sewing integrals involved. For rough weak solutions, we can then prove a generalized It\^o formula. Furthermore, we show that canonical weak solutions are wellposed in the Young case (and equivalent to rough weak solutions), while ill-posed in the rough case. For the latter, we construct a counterexample for uniqueness in law.