Mean value formulas and Liouville theorems for linear second order PDEs

We present several Mean Value formulas for solutions to linear second order PDEs endowed with smooth ''local fundamental solutions''. We then show how these formulas can be used to obtain Liouville Theorems for entire solutions. Our formulas are, in general, weighted average formulas. The relevant weights are ''densities with the mean value property'' a notion playing a central role in rigidity and stability problems. The results we present, related to the Mean value formulas, are obtained in collaboration with Giovanni Cupini. The ones related to the Liouville Theorems are joint works with Alessia Kogoj.