Conjecture on projected Wiener dynamics for mass-conserving Allen–Cahn interfaces

Let $\xi(t)$ be the solution of the stochastic differential equation for the non-mass-conserving interface positions, with coefficients $b$ and $\sigma$ given by the source equations and with $h$ replaced by $\xi$. Let $\bar{\xi}(t)$ be the projection of the Wiener process $W$ onto the mass-conserving manifold $\mathcal{M}_\mu$. For $m>0$ and small $\kappa>0$, define

\tau=\inf\left\\{t\geq0:\xi\notin\mathcal{A}_{\rho_\varepsilon}\text{ or }\lVert v(t)\rVert>\varepsilon^{3/2+m}\text{ or }\lVert v(t)\rVert_{L^4}>\varepsilon^{3/4+m/2-\kappa}\right\\}.

Projected Wiener dynamics conjecture. For $T_\varepsilon\leq c\varepsilon\eta_\varepsilon^{-1}\wedge\tau$, one expects

$$\mathbb{E}\sup_{0\leq t\leq T_\varepsilon}|\xi(t)-\bar{\xi}(t)|\leq c\left[\eta_\varepsilon+\varepsilon^{3+2m-2\kappa}\right]T_\varepsilon.$$

The claim is the mass-conserving analogue of the established approximation for ordinary Allen–Cahn dynamics. The authors expect it to be provable but omit the details; their error estimate does not reach the full relevant time scale because of the weaker spectral gap and the resulting restriction on the noise strength.