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In this paper we find viscosity solutions to an elliptic system governed bytwo different operators (the Laplacian and the infinity Laplacian) using aprobabilistic approach. We analyze a game that combines the Tug-of-War withRandom Walks in two different boars. We show that these value functionsconverge uniformly to a viscosity solution of the elliptic system as the stepsize goes to zero.  In addition, we show uniqueness for the elliptic system using pure PDEtechniques.

A game theoretical approach for an elliptic system with two different operators (the Laplacian and the infinity Laplacian)