Fundamental solution of kinetic Fokker-Planck operator with anisotropic nonlocal dissipativity

TL;DR

This paper proves the existence of smooth fundamental solutions for a degenerate kinetic Fokker-Planck equation with anisotropic nonlocal dissipativity using Malliavin calculus, accommodating anisotropic Lévy processes and cubic growth drifts.

Contribution

It introduces a novel approach to establish fundamental solutions for kinetic Fokker-Planck equations with anisotropic nonlocal dissipativity.

Findings

Existence of smooth fundamental solutions proven

Applicable to equations with anisotropic Lévy process generators

Handles cubic growth drift terms

Abstract

By using the probability approach (the Malliavin calculus), we prove the existence of smooth fundamental solutions for degenerate kinetic Fokker-Planck equation with anisotropic nonlocal dissipativity, where the dissipative term is the generator of an anisotropic L\'evy process and the drift term is allowed to be cubic growth.