Loss without recovery of Gibbsianness during diffusion of continuous spins

TL;DR

This paper investigates the Gibbsian properties of continuous-spin models under diffusion, showing that Gibbsianity can be lost over time without recovery, especially in the presence of external fields, contrasting with discrete models.

Contribution

It demonstrates that continuous-spin Gibbs measures can lose their Gibbsian property over time without recovering, unlike discrete-spin models, in a general dimension setting.

Findings

High temperature initial measures remain Gibbsian for all times.

Low temperature measures are Gibbsian for small times but lose this property at large times.

No Gibbsian recovery occurs for large times in the presence of a magnetic field.

Abstract

We consider a specific continuous-spin Gibbs distribution $\mu_{t=0}$ for a double-well potential that allows for ferromagnetic ordering. We study the time-evolution of this initial measure under independent diffusions. For `high temperature' initial measures we prove that the time-evoved measure $\mu_{t}$ is Gibbsian for all $t$. For `low temperature' initial measures we prove that $\mu_t$ stays Gibbsian for small enough times $t$, but loses its Gibbsian character for large enough $t$. In contrast to the analogous situation for discrete-spin Gibbs measures, there is no recovery of the Gibbs property for large $t$ in the presence of a non-vanishing external magnetic field. All of our results hold for any dimension $d\geq 2$. This example suggests more generally that time-evolved continuous-spin models tend to be non-Gibbsian more easily than their discrete-spin counterparts.