Systems involving mean value formulas on trees
Alfredo Miranda, Carolina A. Mosquera, Julio D. Rossi
TL;DR
This work studies the Dirichlet problem for systems of mean-value equations on regular $m$-ary trees, addressing both directed and undirected variants with linear couplings between components. It derives sharp necessary and sufficient conditions on the level-dependent coefficients $\beta_k^u,\beta_k^v,p_k,q_k$ to guarantee existence and uniqueness of continuous-boundary solutions, and develops a framework (Perron’s method and comparison principles) to handle general averaging operators $F$ and $G$. A concrete directed-tree case with simple averaging yields explicit constant-boundary solutions and clarifies when solvability fails if $\sum_{k} p_k$ diverges; the results are extended to the undirected setting with general $F,G$ and nonzero $\beta$’s, establishing existence/uniqueness under summability conditions. Furthermore, a game-theoretic interpretation links the mean-value dynamics to the limit of value functions of a two-board, zero-sum game, providing an intuitive probabilistic perspective on the coupling and boundary mechanisms.
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.