Random walks and random tug of war in the Heisenberg group

We study the mean value properties of $\mathbf{p}$-harmonic functions on the first Heisenberg group $\mathbb{H}$, in connection to the dynamic programming principles of certain stochastic processes. We implement the approach of Peres-Scheffield to provide the game-theoretical interpretation of the sub-elliptic $\mathbf{p}$-Laplacian; and of Manfredi-Parviainen-Rossi to characterize its viscosity solutions via the asymptotic mean value expansions.