Conjecture on the sharp weak-noise threshold for spatial ergodicity

Let $u_\theta$ denote the annealed invariant measure obtained from the constant initial profile $\theta\ri\in\mathbb{R}$, and let and denote, respectively, the stronger and baseline weak-noise conditions introduced in the paper. Spatial-ergodicity threshold conjecture. The condition is not optimal for ensuring spatial ergodicity of the measures $u_\theta$, while the weaker condition is not sufficient. The paper gives no rigorous argument establishing this claim; determining the sharp threshold remains open.