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 $u\in W_{\mathrm{loc}}^{1,\infty }(\hat{I}\text{c◯})$ is simultaneously a viscosity solution of both of the equations $A_H(u)=f(x)$ and $A_H(u)=g(x)$ in $\hat{I}\text{c◯}$, where $f,g\in C(\hat{I}\text{c◯}),$ then $f=g.$ The assumption $u\in W_{loc}^{1,\infty }(\hat{I}\text{c◯})$ can be relaxed to $u\in C(\hat{I}\text{c◯})$ in many interesting situations. Also, we prove that if $f,g,u\in C(\hat{I}\text{c◯})$ and $u$ is simultaneously a viscosity solution of the equations $\frac{\hat{I}{”}_{\infty }u}{|Du{|}^2}=-f(x)$ and $\frac{\hat{I}{”}_{\infty }u}{|Du{|}^2}=-g(x)$ in $\hat{I}\text{c◯}$ then $f=g.$ This answers a question posed in Peres, Schramm, Scheffield and Wilson [PSSW] concerning whether or not the value function uniquely determines the running cost in the "tug-of-war" game.

To appear in "Ann. Inst. H. Poincare Anal. Non Lineaire"