%0 Journal Article
%T A Game-Tree approach to discrete infinity Laplacian with running costs
%A Qing Liu
%A Armin Schikorra
%J Mathematics
%D 2013
%I arXiv
%X We give a self-contained and elementary proof for boundedness, existence, and uniqueness of solutions to dynamic programming principles (DPP) for biased tug-of-war games with running costs. The domain we work in is very general, and as a special case contains metric spaces. Technically, we introduce game-trees and show that a discretized flow converges uniformly, from which we obtain not only the existence, but also the uniqueness. Our arguments are entirely deterministic, and also do not rely on (semi-)continuity in any way; in particular, we do not need to mollify the DPP at the boundary for well-posedness.
%U http://arxiv.org/abs/1305.7372v2