# Nonlocal isoperimetric inequalities for Kolmogorov-Fokker-Planck operators

## TL;DR

This paper establishes optimal nonlocal isoperimetric inequalities for fractional Kolmogorov-Fokker-Planck operators, which are degenerate and lack variational structure, extending ideas from local cases and physics applications.

## Contribution

It introduces the first optimal isoperimetric inequalities for nonlocal fractional Kolmogorov-Fokker-Planck operators, a class of degenerate, non-variational operators relevant in physics.

## Key findings

- Derived optimal isoperimetric inequalities for fractional Kolmogorov-Fokker-Planck operators
- Extended classical inequalities to a nonlocal, degenerate setting
- Provided insights into the geometric analysis of nonlocal PDEs

## Abstract

In this paper we establish optimal isoperimetric inequalities for a nonlocal perimeter adapted to the fractional powers of a class of Kolmogorov-Fokker-Planck operators which are of interest in physics. These operators are very degenerate and do not possess a variational structure. The prototypical example was introduced by Kolmogorov in his 1938 paper on brownian motion and the theory of gases. Our work has been influenced by ideas of M. Ledoux in the local case.