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

We prove mean value formulas for classical solutions to second order linear differential equations in the for $$\begin{aligned}{ \partial _t u = \sum\nolimits _{i,j=1}^m X_i (a_{ij} X_j u) + X_0 u + \sum\nolimits _{j=1}^m b_j X_j u + cu + f,} \end{aligned}$$ 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 left invariant with respect to a suitable homogeneous Lie group. Our results apply e.g. to degenerate Kolmogorov operators and parabolic equations on Carnot groups $$ \partial _t u = \sum _{i,j=1}^m X_i (a_{ij} X_j u) + \sum _{j=1}^m b_j X_j u + c u + f$$ .