Stationary solutions and local equations for interacting diffusions on regular trees

TL;DR

This paper characterizes invariant measures of infinite interacting diffusions on regular trees using local equations and fixed point analysis, revealing new insights into spectral measures and repulsion effects.

Contribution

It introduces a novel local equation framework for analyzing invariant measures of SDE systems on trees, with existence and uniqueness results and applications to spectral laws and repulsive interactions.

Findings

Derived a new characterization of joint laws at adjacent vertices.

Proved existence and uniqueness of invariant measures for the local equation.

Applied methods to recover the Kesten-McKay law and construct repulsive SDE solutions.

Abstract

We study the invariant measures of infinite systems of stochastic differential equations (SDEs) indexed by the vertices of a regular tree. These invariant measures correspond to Gibbs measures associated with certain continuous specifications, and we focus specifically on those measures which are homogeneous Markov random fields. We characterize the joint law at any two adjacent vertices in terms of a new two-dimensional SDE system, called the "local equation", which exhibits an unusual dependence on a conditional law. Exploiting an alternative characterization in terms of an eigenfunction-type fixed point problem, we derive existence and uniqueness results for invariant measures of the local equation and infinite SDE system. This machinery is put to use in two examples. First, we give a detailed analysis of the surprisingly subtle case of linear coefficients, which yields a new way to derive the famous Kesten-McKay law for the spectral measure of the regular tree. Second, we construct solutions of tree-indexed SDE systems with nearest-neighbor repulsion effects, similar to Dyson's Brownian motion.