Krylov-Safonov theory for Pucci-type extremal inequalities on random data clouds (2410.01642v1)

Abstract: We establish Krylov-Safonov type H\"older regularity theory for solutions to quite general discrete dynamic programming equations or equivalently discrete stochastic processes on random geometric graphs. Such graphs arise for example from data clouds in graph-based machine learning. The results actually hold to functions satisfying Pucci-type extremal inequalities, and thus we cover many examples including tug-of-war games on random geometric graphs. As an application we show that under suitable assumptions when the number of data points increases, the graph functions converge to a solution of a partial differential equation.