Fractional Kolmogorov equations with singular paracontrolled terminal conditions

TL;DR

This paper develops a new solution theory for backward fractional Kolmogorov equations with singular drift and terminal conditions, extending previous work to more singular data using paracontrolled calculus.

Contribution

It introduces a paracontrolled solution space that handles singular terminal conditions without modified paraproducts, generalizing prior results.

Findings

Established a unified solution framework for singular and non-singular data.

Extended the theory to drifts beyond the Young regime.

Provided tools applicable to general linear PDEs with paracontrolled methods.

Abstract

We consider backward fractional Kolmogorov equations with singular Besov drift of low regularity and singular terminal conditions. To treat drifts beyond the socalled Young regime, we assume an enhancement assumption on the drift and consider paracontrolled terminal conditions. Our work generalizes previous results on the equation from Cannizzaro, Chouk 2018 and Kremp, Perkowski 2022 to the case of singular paracontrolled terminal conditions and simultaneously treats singular and non-singular data in one concise solution theory. We introduce a paracontrolled solution space, that implies parabolic time and space regularity on the solution without introducing the socalled "modified paraproduct" from Gubinelli, Perkowski 2017. The tools developed in this article apply for general linear PDEs that can be tackled with the paracontrolled ansatz.