Wong–Zakai Approximations of Stochastic Allen–Cahn Equation
Keywords:
Stochastic Allen–Cahn equation, Wong–Zakai approximations, strong convergence rate.Abstract
We establish an unconditional and optimal strong convergence rate of Wong\u2013Zakai\r type approximations in Banach space norm for a parabolic stochastic partial differential equation\r with monotone drift, including the stochastic Allen\u2013Cahn equation, driven by an additive Brownian sheet. The key ingredient in the analysis is the full use of additive nature of the noise and\r monotonicity of the drift to derive a priori estimation for the solution of this equation. Then\r we use the factorization method and stochastic calculus in martingale type 2 Banach spaces to\r deduce sharp error estimation between the exact and approximate Ornstein\u2013Uhlenbeck processes,\r in Banach space norm. Finally, we combine this error estimation with the aforementioned a priori\r estimation to deduce the desired strong convergence rate of Wong\u2013Zakai type approximations.