Strong convergence conjecture for the stochastic Allen–Cahn equation with infinite-dimensional noise

Consider the stochastic Allen–Cahn equation driven by a multiplicative infinite-dimensional ${\bf Q}$-Wiener process, with the drift-implicit Euler–Galerkin scheme and numerical approximations $u_h^m$ at times $t_m$. Let $h$ denote the spatial mesh size, let $\tau$ denote the time-step size, and assume $u_0\in \dot H^\mathbf s$ for $\mathbf s=1,2$. Strong convergence conjecture. Under mild assumptions on the diffusion coefficients, the scheme satisfies

$$\sup_{0\le m\le M}\|u(t_m)-u_h^m\|_{L_\omega^2 L_x^2}=\mathcal O\left(h^\mathbf s+\tau^{\frac12}\right),\qquad \mathbf s=1,2.$$

This conjecture predicts optimal spatial strong convergence together with the standard one-half order in time for the infinite-dimensional multiplicative-noise case. The corresponding general result under the stated conditions remains unknown and is an open problem.