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Numerical Ergodicity and Uniform Estimate of Monotone SPDEs Driven by Multiplicative Noise

Zhihui Liu

TL;DR

This work analyzes long-time behavior and ergodicity of numerical schemes for monotone SPDEs with multiplicative noise, establishing exponential ergodicity and a unique invariant measure for both time-discrete (DIE) and spatially discrete (DIEG) schemes. It develops uniform in time a priori estimates for the exact solution and transfers these to the numerical schemes, proving exponential mixing and time-independent strong error bounds that quantify convergence to the true solution as mesh size $h$ and time step $\tau$ shrink. The results are specialized to the stochastic Allen–Cahn equation, where the authors show the existence of invariant measures and exponential ergodicity for the continuous and discrete systems, with a threshold on the interface thickness $\alpha$ ensuring ergodicity and, in some cases, uniqueness of the invariant measure. Collectively, the paper advances numerical ergodicity theory for SPDEs under multiplicative noise and provides rigorous, time-uniform error estimates for long-time simulations, with implications for accurate computation of ergodic limits in applied contexts.

Abstract

We analyze the long-time behavior of numerical schemes for a class of monotone stochastic partial differential equations (SPDEs) driven by multiplicative noise. By deriving several time-independent a priori estimates for the numerical solutions, combined with the ergodic theory of Markov processes, we establish the exponential ergodicity of these schemes with a unique invariant measure, respectively. Applying these results to the stochastic Allen--Cahn equation indicates that these schemes always have at least one invariant measure, respectively, and converge strongly to the exact solution with sharp time-independent rates. We also show that these numerical invariant measures are exponentially ergodic and thus give an affirmative answer to a question proposed in (J. Cui, J. Hong, and L. Sun, Stochastic Process. Appl. (2021): 55--93), provided that the interface thickness is not too small.

Numerical Ergodicity and Uniform Estimate of Monotone SPDEs Driven by Multiplicative Noise

TL;DR

This work analyzes long-time behavior and ergodicity of numerical schemes for monotone SPDEs with multiplicative noise, establishing exponential ergodicity and a unique invariant measure for both time-discrete (DIE) and spatially discrete (DIEG) schemes. It develops uniform in time a priori estimates for the exact solution and transfers these to the numerical schemes, proving exponential mixing and time-independent strong error bounds that quantify convergence to the true solution as mesh size and time step shrink. The results are specialized to the stochastic Allen–Cahn equation, where the authors show the existence of invariant measures and exponential ergodicity for the continuous and discrete systems, with a threshold on the interface thickness ensuring ergodicity and, in some cases, uniqueness of the invariant measure. Collectively, the paper advances numerical ergodicity theory for SPDEs under multiplicative noise and provides rigorous, time-uniform error estimates for long-time simulations, with implications for accurate computation of ergodic limits in applied contexts.

Abstract

We analyze the long-time behavior of numerical schemes for a class of monotone stochastic partial differential equations (SPDEs) driven by multiplicative noise. By deriving several time-independent a priori estimates for the numerical solutions, combined with the ergodic theory of Markov processes, we establish the exponential ergodicity of these schemes with a unique invariant measure, respectively. Applying these results to the stochastic Allen--Cahn equation indicates that these schemes always have at least one invariant measure, respectively, and converge strongly to the exact solution with sharp time-independent rates. We also show that these numerical invariant measures are exponentially ergodic and thus give an affirmative answer to a question proposed in (J. Cui, J. Hong, and L. Sun, Stochastic Process. Appl. (2021): 55--93), provided that the interface thickness is not too small.
Paper Structure (10 sections, 11 theorems, 80 equations)

This paper contains 10 sections, 11 theorems, 80 equations.

Table of Contents

  1. Introduction
  2. Uniform A Priori Estimates and Exponential Ergodicity of Monotone SPDEs
  3. Main Assumptions
  4. Uniform Moments' Estimates of The Exact Solution
  5. Exponential Ergodicity of The Exact Solution
  6. Exponential Ergodicity and Uniform Estimates of Galerkin-based Euler Scheme
  7. Exponential Ergodicity of DIE Scheme
  8. Exponential Ergodicity of DIEG Scheme
  9. Uniform Strong Error estimate of DIEG Scheme
  10. Applications to Stochastic Allen--Cahn Equation

Key Result

Proposition 2.1

Let $p \ge 2$, $X_0 \in H$, Assumption ap-f and conditions g-lip--g-gro hold. Assume that $L_2+\frac{p-1}{2} L_7<\lambda_1$. Then there exist positive constants $\gamma_1$ and $C_{\gamma_1}$ such that Moreover, if $X_0 \in \dot H^1$, $L_2+\frac{p-1}{2} L_9 < \lambda_1$, and condition g-gro-1 hold, then there exist positive constants $\gamma_2$ and $C_{\gamma_2}$ such that

Theorems & Definitions (25)

  • Example 2.1
  • Remark 2.1
  • Proposition 2.1
  • proof
  • Corollary 2.1
  • proof
  • Theorem 2.1
  • Remark 2.2
  • Lemma 3.1
  • proof
  • ...and 15 more