Work

Fundamental Solutions to Kolmogorov–Fokker–Planck Equations with Rough Coefficients: Existence, Uniqueness, Upper Estimates

Year: 2025

Type: article

Abstract: . We show the existence and uniqueness of fundamental solution operators to Kolmogorov–Fokker–Planck equations with rough (measurable) coefficients and local or integral diffusion on finite and infinite time strips. In the local case, that is, when the diffusion operator is of differential type, we prove [Formula: see text] decay using Davies’s method and the conservation property. We also prove that the existence of a generalized fundamental solution with the expected pointwise Gaussian upper bound is equivalent to Moser’s [Formula: see text] estimates for local weak solutions to the equation and its adjoint. When coefficients are real, this gives the existence and uniqueness of such a generalized fundamental solution and a new and natural way to obtain pointwise decay.

Source: SIAM Journal on Mathematical Analysis

DOI: 10.1137/24m1649241

Authors: Pascal Auscher, Cyril Imbert, Lukas Niebel