Fractional Kolmogorov Equations with Singular Paracontrolled Terminal Conditions

We consider backward fractional Kolmogorov equations with singular Besov drift of low regularity and singular terminal conditions. To treat drifts beyond the so-called 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 and Chouk (Ann Probab 46:1710-1763, 2018), Kremp and Perkowski (Bernoulli 28:1757-1783, 2022. 10.3150/21-BEJ1394) 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 so-called modified paraproduct from Gubinelli and Perkowski (Commun Math Phys 349:165-269, 2017). The tools developed in this article apply for general linear PDEs that can be tackled with the paracontrolled ansatz.