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Approximation of the invariant measure for stochastic Allen-Cahn equation via an explicit fully discrete scheme

Yibo Wang, Wanrong Cao

TL;DR

This work addresses approximating the invariant measure of the stochastic Allen–Cahn equation driven by a $Q$-Wiener process. It introduces an explicit fully discrete scheme built from a spectral Galerkin spatial discretization and a taming accelerated exponential Euler time integrator, then proves time-uniform weak convergence on infinite horizons via Malliavin calculus. The main results provide a sharp weak error rate of order $\beta$ in time and $\lambda_{N+1}^{-\beta}$ in space, with an overall invariant-measure approximation error of $\mathcal{O}(\lambda_{N+1}^{-\beta}+\tau^{\beta})$ plus an exponential ergodicity term. These findings enable efficient long-time simulations and reliable estimation of the system's invariant measure using explicit discretizations, with stability guaranteed by time-independent moment bounds.

Abstract

In this paper we propose an explicit fully discrete scheme to numerically solve the stochastic Allen-Cahn equation. The spatial discretization is done by a spectral Galerkin method, followed by the temporal discretization by a tamed accelerated exponential Euler scheme. Based on the time-independent boundedness of moments of numerical solutions, we present the weak error analysis in an infinite time interval by using Malliavin calculus. This provides a way to numerically approximate the invariant measure for the stochastic Allen-Cahn equation.

Approximation of the invariant measure for stochastic Allen-Cahn equation via an explicit fully discrete scheme

TL;DR

This work addresses approximating the invariant measure of the stochastic Allen–Cahn equation driven by a -Wiener process. It introduces an explicit fully discrete scheme built from a spectral Galerkin spatial discretization and a taming accelerated exponential Euler time integrator, then proves time-uniform weak convergence on infinite horizons via Malliavin calculus. The main results provide a sharp weak error rate of order in time and in space, with an overall invariant-measure approximation error of plus an exponential ergodicity term. These findings enable efficient long-time simulations and reliable estimation of the system's invariant measure using explicit discretizations, with stability guaranteed by time-independent moment bounds.

Abstract

In this paper we propose an explicit fully discrete scheme to numerically solve the stochastic Allen-Cahn equation. The spatial discretization is done by a spectral Galerkin method, followed by the temporal discretization by a tamed accelerated exponential Euler scheme. Based on the time-independent boundedness of moments of numerical solutions, we present the weak error analysis in an infinite time interval by using Malliavin calculus. This provides a way to numerically approximate the invariant measure for the stochastic Allen-Cahn equation.
Paper Structure (8 sections, 13 theorems, 133 equations)

This paper contains 8 sections, 13 theorems, 133 equations.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Error estimate for the spatial semi-discretization
  4. Error estimate for the full discretization
  5. Boundedness of moments of numerical solutions
  6. Weak convergence rate of the full discretization
  7. Conclusion
  8. Proof of \ref{['Est:w L8']}

Key Result

Theorem 1.1

Let Assumptions Asp:Initial value--Asp:additional asp hold and $\|A^{\frac{\beta-1}{2}}Q^{\frac{1}{2}}\|_{\mathcal{L}_2(H)}<\infty$, $\beta\in(0,1]$. For test function $\Phi\in C_b^2(H;\mathbb{R})$, $K\geq2$, and $\tau\in(0,\tau_0]$, $\tau_0>0$ is an arbitrary parameter, it holds that where $V_K$ is the numerical solution given by scheme1 and the constant $C>0$ is independent of $N$, $\tau$ and $

Theorems & Definitions (24)

  • Theorem 1.1
  • Corollary 1.2
  • Lemma 2.1
  • Theorem 2.2
  • Theorem 2.3: Proposition 3.3 in Brehier2022
  • Remark 2.4
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • Lemma 3.3
  • ...and 14 more