# American Institute of Mathematical Sciences

May  2017, 16(3): 915-944. doi: 10.3934/cpaa.2017044

## Tug-of-war games with varying probabilities and the normalized p(x)-laplacian

 1 Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Spain 2 Department of Mathematics and Statistics, University of Jyväskylä, PO Box 35, FI-40014 Jyväskylä, Finland

Received  August 2016 Revised  December 2016 Published  February 2017

We study a two player zero-sum tug-of-war game with varying probabilities that depend on the game location x. In particular, we show that the value of the game is locally asymptotically Hölder continuous. The main difficulty is the loss of translation invariance. We also show the existence and uniqueness of values of the game. As an application, we prove that the value function of the game converges to a viscosity solution of the normalized p(x) -Laplacian.

Citation: Ángel Arroyo, Joonas Heino, Mikko Parviainen. Tug-of-war games with varying probabilities and the normalized p(x)-laplacian. Communications on Pure & Applied Analysis, 2017, 16 (3) : 915-944. doi: 10.3934/cpaa.2017044
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