Wong–Zakai Approximations of Stochastic Allen–Cahn Equation

Authors

  • Zhihui Liu Department of Mathematics, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong
  • Zhonghua Qiao Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong

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.

Published

2019-08-09

Issue

Section

Articles