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

Abstract

Let $A_H$ be the Aronsson operator associated with a Hamiltonian $H(x,z,p)$. Aronsson operators arise from $L^{\infty }$ variational problems, two person game theory, control problems, etc. In this paper, we prove, under suitable conditions, that if $u\in W_{\mathrm{loc}}^{1,\infty }(\Omega )$ is simultaneously a viscosity solution of both of the equations

where $f,g\in C(\Omega )$, then $f=g$. The assumption $u\in W_{\mathrm{loc}}^{1,\infty }(\Omega )$ can be relaxed to $u\in C(\Omega )$ in many interesting situations. Also, we prove that if $f,g,u\in C(\Omega )$ and $u$ is simultaneously a viscosity solution of the equations

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.