Uniqueness of values of Aronsson operators and running costs in “tug-of-war” games

Let AHAH be the Aronsson operator associated with a Hamiltonian H(x,z,p)H(x,z,p). Aronsson operators arise from L∞L∞ variational problems, two person game theory, control problems, etc. In this paper, we prove, under suitable conditions, that if u∈Wloc1,∞(Ω) is simultaneously a viscosity solution of both of the equationsequation(0.1)AH(u)=f(x)andAH(u)=g(x)in Ω, where f,g∈C(Ω)f,g∈C(Ω), then f=gf=g. The assumption u∈Wloc1,∞(Ω) can be relaxed to u∈C(Ω)u∈C(Ω) in many interesting situations. Also, we prove that if f,g,u∈C(Ω)f,g,u∈C(Ω) and u is simultaneously a viscosity solution of the equationsequation(0.2)Δ∞u|Du|2=−f(x)andΔ∞u|Du|2=−g(x)in Ω, then f=gf=g. This answers a question posed in Peres, Schramm, Scheffield and Wilson [Y. Peres, O. Schramm, S. Sheffield, D.B. Wilson, Tug-of-war and the infinity Laplacian, J. Amer. Math. Soc. Math. 22 (2009) 167–210] concerning whether or not the value function uniquely determines the running cost in the “tug-of-war” game.