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

Abstract

For any integer $\hat{I}º\ge 2$, the $\hat{I}º$-local-field equation ($\hat{I}º$-LFE) characterizes the limit of the neighborhood path empirical measure of interacting diffusions on $\hat{I}º$-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) $\hat{I}º$-LFE coincides with that of a certain more tractable Markovian analog, the Markov $\hat{I}º$-local-field equation. In the present article, we prove this conjecture for the case when $\hat{I}º=2$ 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 $n\to \infty$ and $t\to \infty$ commute. Along the way, we also establish well-posedness of the Markov $\hat{I}º$-local field equations with affine drifts for all $\hat{I}º\ge 2$, which may be of independent interest.

No changes, other than adding funding acknowledgement