Mean value formulas for Ornstein-Uhlenbeck and Hermite temperatures (Q2188359)

In this article the authors obtain explicit mean value formulae for the solutions of the diffusion equations associated with the Ornstein-Uhlenbeck equation, namely \(U_t-(\Delta-2x\cdot \nabla)U=0\) in \(E\), and Hermite equation, namely \(U_t-(\Delta-|x|^2)U=0\) in \(E,\) where \(E\subset \mathbb{R}^n\times\mathbb{R}\) is an open set. These formulae are further used to derive various useful properties such as maximum principle, uniquenes theorems and Harnack-type inequalities.