Mean value formulas for classical solutions to subelliptic evolution equations in stratified Lie groups

Abstract

We prove mean value formulas for classical solutions to second order linear differential equations in the form $$ \partial_t u = \sum_{i,j=1}^m X_i (a_{ij} X_j u) + X_0 u + f, $$ where $A = (a_{ij})_{i,j=1, \dots,m}$ is a bounded, symmetric and uniformly positive matrix with $C^1$ coefficients under the assumption that the operator $\sum_{j=1}^m X_j^2 + X_0 - \partial_t$ is hypoelliptic and the vector fields $X_1, \dots, X_m$ and $X_{m+1} :=X_0 - \partial_t$ are invariant with respect to a suitable homogeneous Lie group. Our results apply e.g. to degenerate Kolmogorov operators and parabolic equations on Carnot groups $\partial_t u = \sum_{i,j=1}^m X_i (a_{ij} X_j u) + f$.