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

  • Yifeng Yu

    Department of Mathematics, University of California at Irvine, 340 Rowland Hall, Irvine, CA, USA

Abstract

Let be the Aronsson operator associated with a Hamiltonian . Aronsson operators arise from variational problems, two person game theory, control problems, etc. In this paper, we prove, under suitable conditions, that if is simultaneously a viscosity solution of both of the equations

where , then . The assumption can be relaxed to in many interesting situations. Also, we prove that if and is simultaneously a viscosity solution of the equations

then . 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.

Cite this article

Yifeng Yu, Uniqueness of values of Aronsson operators and running costs in “tug-of-war” games. Ann. Inst. H. Poincaré Anal. Non Linéaire 26 (2009), no. 4, pp. 1299–1308

DOI 10.1016/J.ANIHPC.2008.11.001