Well-posedness of Kolmogorov-Fokker-Planck equations with unbounded drift

TL;DR

This paper establishes existence, uniqueness, and stochastic equivalence for Kolmogorov-Fokker-Planck equations with unbounded, measurable-in-time, locally Hölder continuous-in-space drift terms, extending the parametrix method.

Contribution

It extends the parametrix method to handle unbounded drifts in Kolmogorov-Fokker-Planck equations, proving well-posedness under minimal regularity assumptions.

Findings

Proved existence and uniqueness of measure solutions.

Established the equivalence with stochastic formulations.

Extended the parametrix method to unbounded drift settings.

Abstract

We consider Kolmogorov-Fokker-Planck equations with unbounded drift terms which are only measurable in time and locally H\"older continuous in space. In particular, we extend the parametrix method to this setting and we prove existence and uniqueness of measure solutions to the associated Cauchy problem, as well as the equivalence with the corresponding stochastic formulation.