Nonlocal PDEs on Graphs: From Tug-of-War Games to Unified Interpolation on Images and Point Clouds

In this paper, we introduce a new general class of partial difference operators on graphs, which interpolate between the nonlocal ∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\infty $$\end{document}-Laplacian, the Laplacian, and a family of discrete gradient operators. In this context, we investigate an associated Dirichlet problem for this general class of operators and prove the existence and uniqueness of respective solutions. We show that a certain partial difference equation based on this class of operators recovers many variants of a stochastic game known as ‘Tug-of-War’ and extends them to a nonlocal setting. Furthermore, we discuss a connection with certain nonlocal partial differential equations. Finally, we propose to use this class of operators as general framework to solve many interpolation problems in a unified manner as arising, e.g., in image and point cloud processing.