[Paper Review] Wong--Zakai Approximations of Stochastic Allen-Cahn Equation

Zhihui Liu, Zhonghua Qiao|arXiv (Cornell University)|Oct 26, 2017

Stochastic processes and financial applications7 references3 citations

TL;DR

This paper establishes an unconditional and optimal strong convergence rate for Wong–Zakai–Galerkin approximations of the stochastic Allen–Cahn equation driven by an additive space-time Brownian sheet. By exploiting the additive noise structure and drift monotonicity, the authors derive sharp error estimates using factorization and stochastic calculus in martingale type 2 Banach spaces, achieving a convergence rate of $\mathcal{O}\left(\left(\frac{1}{m}\right)^{1/4} \wedge \left(\frac{1}{n}\right)^{1/2}\right)$ in $L^p$-norm for the solution and its approximation.

ABSTRACT

We establish a 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 fully use of additive nature of the noise and monotonicity of the drift to derive a priori estimation for the solution of this equation, in combination with the factorization method and stochastic calculus in martingale type 2 Banach spaces applied to deduce sharp error estimation between the exact and approximate Ornstein--Uhlenbeck processes, in Banach space norm.

Motivation & Objective

• To derive an unconditional and optimal strong convergence rate for Wong–Zakai–Galerkin approximations of the stochastic Allen–Cahn equation with additive space-time white noise.
• To overcome the limitations of conditional convergence rates in prior works by removing probabilistic assumptions on the sample path set.
• To establish sharp error bounds in $L^p$-norms for both the solution and associated Ornstein–Uhlenbeck processes under minimal initial data assumptions.
• To generalize existing convergence results to non-Lipschitz, polynomially growing drifts via novel a priori estimates in Banach space settings.

Proposed method

• Transform the SPDE into an equivalent random PDE using the additive noise structure, enabling separation of the stochastic and deterministic components.
• Apply the factorization method and stochastic calculus in martingale type 2 Banach spaces to derive uniform bounds on the exact and approximate Ornstein–Uhlenbeck processes.
• Use spectral Galerkin projection combined with temporal piecewise constant approximation of the noise to construct the Wong–Zakai–Galerkin scheme.
• Employ the monotonicity of the drift $f$ and polynomial growth conditions to derive a priori estimates for the solution and error terms.
• Apply Grönwall’s inequality and Hölder/Young inequalities to control nonlinear terms arising from the drift in $L^p$-norms.
• Establish convergence rates in $L^ ho(0,T;L^ ho( ho;L^ ho(0,1)))$-type norms for $\rho > 2$, crucial for handling non-Lipschitz nonlinearities.

Experimental results

Research questions

• RQ1 Can an unconditional and optimal strong convergence rate be established for Wong–Zakai approximations of the stochastic Allen–Cahn equation with additive noise?
• RQ2 How does the convergence rate depend on temporal ($m$) and spatial ($n$) discretization parameters under minimal regularity assumptions on the initial data?
• RQ3 What is the sharp error estimate between the exact solution and the Wong–Zakai–Galerkin approximation in Banach space norms for non-Lipschitz drifts?
• RQ4 To what extent can the factorization method and stochastic calculus in martingale type 2 spaces improve convergence analysis for SPDEs with monotone, polynomially growing drifts?
• RQ5 Can the convergence rate be improved beyond conditional estimates that depend on shrinking probability sets?

Key findings

• The paper establishes an unconditional strong convergence rate of $\mathcal{O}\left(\left(\frac{1}{m}\right)^{1/4} \wedge \left(\frac{1}{n}\right)^{1/2}\right)$ in the $L^p(0,T;L^p(\Omega;L^p(0,1)))$-norm for the solution and its approximation, valid for any $1 \leq p < \frac{p_*}{2} + 1$.
• The convergence rate is optimal and holds without conditioning on sample paths, unlike prior results that required $\mathbb{P}(\Omega_{\tau,h}) \to 1$.
• The analysis achieves sharp error estimates by combining factorization techniques with stochastic calculus in martingale type 2 Banach spaces, enabling control of high-order moments.
• For $p \geq 2$, a refined estimate yields $\mathbb{E}\left[\|u - u^{m,n}\|_{L^p}^p\right] + \int_0^T \mathbb{E}\left[\|u - u^{m,n}\|_{L^{p+q-2}}^{p+q-2}\right] dt \leq C\left(1 + \mathbb{E}[\|u_0\|_{L^p}^p]\right)\left(\left(\frac{1}{m}\right)^{1/4} \wedge \left(\frac{1}{n}\right)^{1/2}\right)^{\frac{p+q-2}{q-1}}$, with $q > 2$.
• The initial data requirement is minimal: $u_0 \in L^{p_*}(\Omega; L^{p_*}(0,1))$ suffices for the convergence rate to hold.
• The method generalizes to SPDEs with monotone, polynomially growing drifts and extends beyond Lipschitz nonlinearities, overcoming limitations of prior approaches relying on Lipschitz conditions.