Degenerate Fokker-Planck Equations : Bismut Formula, Gradient Estimate and Harnack Inequality

TL;DR

This paper develops coupling methods for degenerate Fokker-Planck equations, deriving explicit derivative formulas, Harnack inequalities, and applications to gradient estimates, entropy, and heat kernel inequalities.

Contribution

It introduces new coupling techniques for degenerate diffusions, enabling explicit formulas and inequalities for solutions to these equations.

Findings

Derived explicit derivative formulas for degenerate Fokker-Planck solutions

Established Harnack inequalities for degenerate diffusions

Applied results to gradient estimates and entropy inequalities

Abstract

By constructing successful couplings for degenerate diffusion processes, explicit derivative formula and Harnack type inequalities are presented for solutions to a class of degenerate Fokker-Planck equations on $\R^m\times\R^{d}$. The main results are also applied to the study of gradient estimate, entropy/transportation-cost inequality and heat kernel inequalities.