Fractional Kolmogorov Equations with Singular Paracontrolled Terminal Conditions.

Area of Science:

• Stochastic Partial Differential Equations
• Fractional Calculus
• Nonlinear Analysis

Background:

• Existing research on backward fractional Kolmogorov equations often requires strong regularity assumptions on drift and terminal conditions.
• Previous works, such as Cannizzaro and Chouk (2018) and Kremp and Perkowski (2022), addressed specific cases but lacked a unified theory for singular data.
• The 'Young regime' and 'modified paraproduct' are limitations in current analytical tools for these equations.

Purpose of the Study:

• To develop a unified solution theory for backward fractional Kolmogorov equations with singular Besov drift and singular terminal conditions.
• To extend the applicability of existing methods to drifts beyond the 'Young regime'.
• To introduce a new paracontrolled solution space that inherently provides parabolic regularity.

Main Methods:

• Introduction of an enhancement assumption on the drift to handle low regularity.
• Utilization of paracontrolled terminal conditions.
• Development of a new paracontrolled solution space that avoids the need for the 'modified paraproduct'.

Main Results:

• A generalized solution theory is established that encompasses both singular and non-singular data within a single framework.
• The proposed paracontrolled solution space yields parabolic time and space regularity for the solutions.
• The methodology successfully treats drifts beyond the 'Young regime', overcoming limitations of previous studies.

Conclusions:

• The study provides a significant advancement in the analysis of backward fractional Kolmogorov equations with challenging data singularities.
• The developed paracontrolled ansatz and solution space offer a powerful and versatile tool applicable to a broader class of linear partial differential equations.
• This work paves the way for future research into more complex fractional dynamics and related PDEs.