  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.
