TY - JOUR
T1 - Wong–Zakai Approximations of Stochastic Allen–Cahn Equation
AU - Liu , Zhihui
AU - Qiao , Zhonghua
JO - International Journal of Numerical Analysis and Modeling
VL - 5
SP - 681
EP - 694
PY - 2019
DA - 2019/08
SN - 16
DO - http://doi.org/
UR - https://global-sci.org/intro/article_detail/ijnam/13248.html
KW - Stochastic Allen–Cahn equation, Wong–Zakai approximations, strong convergence
rate.
AB - <p style="text-align: justify;">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.</p>