A case study on the long-time behavior of the Gaussian local-field equation

For any integer κ≥2, the κ-local-field equation characterizes the limit of the neighborhood path empirical measure of interacting diffusions on κ-regular random graphs, as the graph size goes to infinity. It has been conjectured that the long-time behavior of the (in general non-Markovian) κ-local field equation coincides with that of a certain more tractable Markovian analog, the Markov κ-local-field equation. In the present article, we prove this conjecture for the case when κ=2 and the diffusions are one-dimensional with affine drifts. As a by-product of our proof, we also show that for interacting diffusions on the n-cycle (or 2-regular random graph on n vertices), the limits n→∞ and t→∞ commute. Along the way, we also establish well-posedness of the Markov κ-local field equations with affine drifts for all κ≥2, which may be of independent interest.