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Numerical Ergodicity of Stochastic Allen--Cahn Equation driven by Multiplicative White Noise

Zhihui Liu

Abstract

We establish the unique ergodicity of a fully discrete scheme for monotone SPDEs with polynomial growth drift and bounded diffusion coefficients driven by multiplicative white noise. The main ingredient of our method depends on the satisfaction of a Lyapunov condition followed by a uniform moments' estimate, combined with the regularity property for the full discretization. We transform the original stochastic equation into an equivalent random equation where the discrete stochastic convolutions are uniformly controlled to derive the desired uniform moments' estimate. Applying the main result to the stochastic Allen--Cahn equation driven by multiplicative white noise indicates that this full discretization is uniquely ergodic for any interface thickness. Numerical experiments validate our theoretical results.

Numerical Ergodicity of Stochastic Allen--Cahn Equation driven by Multiplicative White Noise

Abstract

We establish the unique ergodicity of a fully discrete scheme for monotone SPDEs with polynomial growth drift and bounded diffusion coefficients driven by multiplicative white noise. The main ingredient of our method depends on the satisfaction of a Lyapunov condition followed by a uniform moments' estimate, combined with the regularity property for the full discretization. We transform the original stochastic equation into an equivalent random equation where the discrete stochastic convolutions are uniformly controlled to derive the desired uniform moments' estimate. Applying the main result to the stochastic Allen--Cahn equation driven by multiplicative white noise indicates that this full discretization is uniquely ergodic for any interface thickness. Numerical experiments validate our theoretical results.
Paper Structure (9 sections, 6 theorems, 44 equations, 1 figure)

This paper contains 9 sections, 6 theorems, 44 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Main Assumptions
  4. DIEG scheme and solvability
  5. Preliminaries on Ergodicity of Markov Chains
  6. Unique Ergodicity of DIEG
  7. Lyapunov structure of DIEG
  8. Regularity Property
  9. Numerical Experiments

Key Result

Lemma 2.1

Under the condition f-mon with $(K_1-\lambda_1) \tau<1$, $\hat{F}: V_N \to V_N$ defined in hatF is bijective so that the DIEG scheme die-g can be uniquely solved pathwise. Morevoer, $\hat{F}$ is an open map, i.e. for each open set $A\in \mathcal{B}(V_N)$, $\hat{F}(A)$ is also an open set in $\mathca

Figures (1)

  • Figure 1: Time averages of \ref{['die-g']} for Eq. \ref{['ac']}

Theorems & Definitions (15)

  • Remark 2.1
  • Lemma 2.1
  • proof
  • Lemma 3.1
  • proof
  • Proposition 3.1
  • proof
  • Remark 3.1
  • Remark 3.2
  • Proposition 3.2
  • ...and 5 more