Consistency of tug-of-war type operators on random data clouds

TL;DR

This paper investigates the convergence properties of a tug-of-war operator on geometric graphs constructed from random data clouds, establishing its consistency and connection to a related model problem as data size grows.

Contribution

It introduces a novel analysis of a tug-of-war operator on random data clouds, demonstrating its convergence and consistency in the context of geometric graphs.

Findings

Proves convergence of the value functions as data points increase.

Establishes the operator's consistency on random data clouds.

Connects the tug-of-war operator to a related model problem.

Abstract

In this paper, we study a tug-of-war type operator on geometric graphs and its associated Dirichlet problem on a random data cloud. Specifically, we analyze the convergence of the value functions as the number of data points increases and the step size of the game shrinks. This analysis reveals the connection between our tug-of-war type operator and the corresponding model problem. A key ingredient in establishing this result is the consistency of the operator.