TUG-OF-WAR GAMES WITH VARYING PROBABILITIES AND THE NORMALIZED p(x)-LAPLACIAN

摘要

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 Holder 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.