Volume 16, Issue 5
Wong–Zakai Approximations of Stochastic Allen–Cahn Equation

Zhihui Liu & Zhonghua Qiao

DOI:

Int. J. Numer. Anal. Mod., 16 (2019), pp. 681-694.

Published online: 2019-08

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  • Abstract

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.

  • Keywords

Stochastic Allen–Cahn equation, Wong–Zakai approximations, strong convergence rate.

  • AMS Subject Headings

60H35, 65M12, 65M15

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

zhliu@ust.hk (Zhihui Liu)

zqiao@polyu.edu.hk (Zhonghua Qiao)

  • BibTex
  • RIS
  • TXT
@Article{IJNAM-16-681, author = {Liu , Zhihui and Qiao , Zhonghua }, title = {Wong–Zakai Approximations of Stochastic Allen–Cahn Equation}, journal = {International Journal of Numerical Analysis and Modeling}, year = {2019}, volume = {16}, number = {5}, pages = {681--694}, abstract = {

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.

}, issn = {2617-8710}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/ijnam/13248.html} }
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 -

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.

Zhihui Liu & Zhonghua Qiao. (2019). Wong–Zakai Approximations of Stochastic Allen–Cahn Equation. International Journal of Numerical Analysis and Modeling. 16 (5). 681-694. doi:
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