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

Let A(H) be the Aronsson operator associated with a Hamiltonian H (x, z, p). Aronsson operators arise from L-infinity variational problems, two person game theory, control problems, etc. In this paper, we prove, under suitable conditions, that if u is an element of W-loc(1,infinity) (Omega) is simultaneously a viscosity solution of both of the equations A(H)(u) = f(x) and A(H)(u) = g(x) in Omega, (0.1) where f, g is an element of C(Omega), then f = g. The assumption u is an element of W-loc(1,infinity) (Omega) can be relaxed to u is an element of C(Omega) in many interesting situations. Also, we prove that if f, g, u is an element of C(Omega) and u is simultaneously a viscosity solution of the equations Delta infinity u/|Du|(2) = -f(x) and Delta infinity u/|Du|(2) = -g(x) in Omega, (0.2) then f = 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. (C) 2008 Elsevier Masson SAS. All rights reserved.