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

TL;DR

This paper proves the uniqueness of the Hamiltonian and running costs in Aronsson equations and tug-of-war games, showing that the value function uniquely determines the cost functions under certain conditions.

Contribution

It establishes that simultaneous solutions of Aronsson equations with different right-hand sides imply those sides are equal, addressing a key question in game theory and control problems.

Findings

Uniqueness of the right-hand side functions in Aronsson equations.

Extension of results to continuous solutions.

Resolution of a question in tug-of-war game theory.

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 $u\in W^{1,\infty}_{\rm loc}(\Omega)$ is simultaneously a viscosity solution of both of the equations $A_H(u)=f(x)$ and $A_H(u)=g(x)$ in $\Omega$, where $f, g\in C(\Omega),$ then $f=g.$ The assumption $u\in W_{loc}^{1,\infty}(\Omega)$ 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 ${\Delta_\infty u\over |Du|^2}=-f(x)$ and ${\Delta_{\infty}u\over |Du|^2}=-g(x)$ in $\Omega$ 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.