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The stochastic Allen–Cahn equation with multiplicative noise involves the nonlinear drift operator  ⁢   (  x  )    =    Δ  ⁢  x   -    (|  x  |   2   -  1   )   ⁢  x{{\mathscr{A}}(x)=\Delta x-(|x|^{2}-1)x}   . We use the fact that  ⁢   (  x  )    =   -      ′   ⁢   (  x  ){{\mathscr{A}}(x)=-{\mathcal{J}}^{\prime}(x)}    satisfies a weak monotonicity property to deduce uniform bounds in strong norms for solutions of the temporal, as well as of the spatio-temporal discretization of the problem. This weak monotonicity property then allows for the estimate

Optimal Strong Rates of Convergence for a Space-Time Discretization of the Stochastic Allen–Cahn Equation with Multiplicative Noise