Sobolev-norm limit conjecture for regularized stochastic Allen–Cahn solutions

Let $u_N$ be the solution to the regularized problem, and let $\|\cdot\|_s$ denote the $H^s(\mathbb{T}^2)$ norm. For every $t>0$, the Sobolev-norm limit conjecture.

$$\lim_{N\to\infty}\mathbb{E}\|u_N(t)\|_s^2= \begin{cases} +\infty & \text{if }s\geq 0,\\ 0 & \text{if }s<0. \end{cases}$$

This predicts simultaneous divergence in nonnegative Sobolev norms and convergence to zero in the mean-square negative Sobolev scale, complementing the zero-distribution conjecture. The claim is motivated by Fourier-mode estimates and is not resolved in the source.