Mean value formulas for Ornstein-Uhlenbeck and Hermite temperatures

TL;DR

This paper derives explicit mean value formulas for solutions to diffusion equations linked with Ornstein-Uhlenbeck and Hermite operators, leading to important properties like maximum principles and Harnack inequalities.

Contribution

It introduces explicit mean value formulas for these diffusion equations, enabling new proofs of classical properties and inequalities.

Findings

Derived explicit mean value formulas for Ornstein-Uhlenbeck and Hermite equations.

Established maximum principles and uniqueness theorems.

Proved Harnack-type inequalities for solutions.

Abstract

We obtain explicit mean value formulas for the solutions of the diffusion equations associated with the Ornstein-Uhlenbeck and Hermite operators. From these, we derive various useful properties, such as maximum principles, uniqueness theorems and Harnack-type inequalities.