# Fractional Kolmogorov Equations with Singular Paracontrolled Terminal Conditions

## TL;DR

This paper extends the theory of fractional Kolmogorov equations to handle singular terminal conditions and low-regularity drifts using paracontrolled methods.

## Contribution

The novel contribution is a unified solution theory for singular and non-singular data using paracontrolled terminal conditions without requiring modified paraproducts.

## Key findings

- A paracontrolled solution space is introduced that ensures regularity without modified paraproducts.
- The approach generalizes previous results to singular paracontrolled terminal conditions.
- The method applies broadly to linear PDEs using the paracontrolled ansatz.

## Abstract

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.