# Global Hypoellipticity and Compactness of Resolvent for Fokker-Planck Operator

## TL;DR

This paper investigates conditions on the potential in the Fokker-Planck operator to ensure global hypoellipticity and resolvent compactness, improving previous results with weaker assumptions.

## Contribution

It provides new global hypoelliptic estimates for the Fokker-Planck operator under less restrictive conditions on the potential.

## Key findings

- Established conditions for hypoellipticity of the Fokker-Planck operator.
- Proved compactness of the resolvent under certain potential behaviors.
- Improved upon previous results by weakening assumptions on the potential.

## Abstract

In this paper we study the Fokker-Planck operator with potential V(x), and analyze some kind of conditions imposed on the potential to ensure the validity of global hypoelliptic estimates. As a consequence, we obtain the compactness of resolvent of the Fokker-Planck operator if either the Witten Laplacian on 0-forms has a compact resolvent or some additional assumption on the behavior of the potential at infinity is fulfilled. This work improves the previous results of H\'erau-Nier and Helffer-Nier, by obtaining a better global hypoelliptic estimate under weaker assumptions on the potential.