Wong–Zakai approximations of stochastic Allen–Cahn equation

We establish an unconditional and optimal strong convergence rate of Wong–Zakai type approximations in Banach space norm for a parabolic stochastic partial differential equation with monotone drift, including the stochastic Allen–Cahn equation, driven by an additive Brownian sheet. The key ingredient in the analysis is the full use of additive nature of the noise and monotonicity of the drift to derive a priori estimation for the solution of this equation. Then we use the factorization method and stochastic calculus in martingale type 2 Banach spaces to deduce sharp error estimation between the exact and approximate Ornstein–Uhlenbeck processes, in Banach space norm. Finally, we combine this error estimation with the aforementioned a priori estimation to deduce the desired strong convergence rate of Wong–Zakai type approximations.