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

Ángel Arroyo; Joonas Heino; Mikko Parviainen

doi:10.3934/cpaa.2017044

Article Contents

2017, Volume 16, Issue 3: 915-944. Doi: 10.3934/cpaa.2017044

This issue Previous Article Weighted lorentz estimates for nondivergence linear elliptic equations with partially BMO coefficients Next Article Non-collapsing for a fully nonlinear inverse curvature flow

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: July 31, 2016

Revised: November 30, 2016

Published: January 2018

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Abstract

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.

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Mathematics Subject Classification: 35J60, 35J92, 35B65, 91A15.

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