Zero-distribution conjecture for the regularized two-dimensional stochastic Allen–Cahn equation

Let $u_N$ be the regularizations of the two-dimensional stochastic Allen–Cahn equation, and let $\langle\cdot,\cdot\rangle$ denote the $L^2$-duality pairing. For every $t>0$ and every smooth test function $\phi$, the zero-distribution conjecture.

$$\lim_{N\to\infty}\left\langle u_N(t),\phi\right\rangle=0$$

in probability. This conjecture makes precise the observed loss of macroscopic structure in the regularized fields: their distributional limit is expected to be zero, even though their pointwise or $L^2$ behavior becomes increasingly oscillatory. The source gives heuristic and numerical support but no proof or resolution.