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MiS Preprint

A Game-Tree approach to discrete infinity Laplacian with running costs

Qing Liu and Armin Schikorra


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.

MSC Codes:
35A35, 49C20, 91A05

Related publications

2013 Repository Open Access
Qing Liu and Armin Schikorra

A Game-Tree approach to discrete infinity Laplacian with running costs