Open Access
2023 Stationary solutions and local equations for interacting diffusions on regular trees
Daniel Lacker, Jiacheng Zhang
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Electron. J. Probab. 28: 1-37 (2023). DOI: 10.1214/22-EJP889


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.


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Daniel Lacker. Jiacheng Zhang. "Stationary solutions and local equations for interacting diffusions on regular trees." Electron. J. Probab. 28 1 - 37, 2023.


Received: 4 December 2021; Accepted: 7 December 2022; Published: 2023
First available in Project Euclid: 4 January 2023

Digital Object Identifier: 10.1214/22-EJP889

Primary: 60G10 , 60K35

Keywords: Gibbs measures , Interacting diffusions , Kesten-McKay law , local equations , Markov random fields , nonlinear Markov processes , regular trees , Repulsive Particle Systems , sparse graphs

Vol.28 • 2023
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