Systems involving mean value formulas on trees

TL;DR

This paper investigates the Dirichlet problem for systems of mean value equations on regular trees, establishing conditions for solutions and connecting them to game theory in specific cases.

Contribution

It provides necessary and sufficient conditions for existence and uniqueness of solutions to mean value systems on trees, including a game-theoretic interpretation.

Findings

Conditions for solution existence and uniqueness

Extension to directed and undirected cases

Connection to zero-sum game limits

Abstract

In this paper we study the Dirichlet problem for systems of mean value equations on a regular tree. We deal both with the directed case (the equations verified by the components of the system at a node in the tree only involve values of the unknowns at the successors of the node in the tree) and the undirected case (now the equations also involve the predecessor in the tree). We find necessary and sufficient conditions on the coefficients in order to have existence and uniqueness of solutions for continuous boundary data. In a particular case, we also include an interpretation of such solutions as a limit of value functions of suitable two-players zero-sum games.