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An inverse mean value property for evolution equations

The aim of this work is to extend a result by Suzuki and Watson concerning an inverse property for caloric functions. Our result applies, in particular, to the heat operator on stratied Lie groups and to Kolmogorov-Fokker-Planck-type operators. We show that the open sets characterizing the solutions to the involved equations, in terms of suitable average operators, have to be the level sets of the fundamental solutions of the relevant operators. The technique adopted exploits the structure of the propagation sets, i.e., the sets where the solutions to the involved equations attain their maximum.