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Numerical Unique Ergodicity of Monotone SDEs driven by Nondegenerate Multiplicative Noise

Zhihui Liu, Zhizhou Liu

TL;DR

This work addresses numerical ergodicity for monotone SDEs and SPDEs driven by nondegenerate multiplicative noise. It develops a Lyapunov–drift framework and a minorization condition to prove the unique ergodicity of the stochastic theta method (STM) for $\theta\in[1/2,1]$ and extends the approach to Galerkin-based full discretizations (DIEG) for SPDEs, with applications to stochastic Allen–Cahn. The results establish geometric ergodicity for $\theta\in(1/2,1]$ and unique ergodicity at $\theta=1/2$, under explicit conditions on step size and coefficients. Numerical experiments on both STM and DIEG corroborate the theory, demonstrating convergence of time-averages to a single invariant measure and validating the practical applicability of the proposed discretizations.

Abstract

We first establish the unique ergodicity of the stochastic theta method (STM) with $θ\in [1/2, 1]$ for monotone SODEs, without growth restriction on the coefficients, driven by nondegenerate multiplicative noise. The main ingredient of the arguments lies in constructing new Lyapunov functions involving the coefficients, the stepsize, and $θ$ and deriving a minorization condition for the STM. We then generalize the arguments to the Galerkin-based full discretizations for a class of monotone SPDEs driven by infinite-dimensional nondegenerate multiplicative trace-class noise. Applying these results to the stochastic Allen--Cahn equation indicates that its Galerkin-based full discretizations are uniquely ergodic for any interface thickness. Numerical experiments verify our theoretical results.

Numerical Unique Ergodicity of Monotone SDEs driven by Nondegenerate Multiplicative Noise

TL;DR

This work addresses numerical ergodicity for monotone SDEs and SPDEs driven by nondegenerate multiplicative noise. It develops a Lyapunov–drift framework and a minorization condition to prove the unique ergodicity of the stochastic theta method (STM) for and extends the approach to Galerkin-based full discretizations (DIEG) for SPDEs, with applications to stochastic Allen–Cahn. The results establish geometric ergodicity for and unique ergodicity at , under explicit conditions on step size and coefficients. Numerical experiments on both STM and DIEG corroborate the theory, demonstrating convergence of time-averages to a single invariant measure and validating the practical applicability of the proposed discretizations.

Abstract

We first establish the unique ergodicity of the stochastic theta method (STM) with for monotone SODEs, without growth restriction on the coefficients, driven by nondegenerate multiplicative noise. The main ingredient of the arguments lies in constructing new Lyapunov functions involving the coefficients, the stepsize, and and deriving a minorization condition for the STM. We then generalize the arguments to the Galerkin-based full discretizations for a class of monotone SPDEs driven by infinite-dimensional nondegenerate multiplicative trace-class noise. Applying these results to the stochastic Allen--Cahn equation indicates that its Galerkin-based full discretizations are uniquely ergodic for any interface thickness. Numerical experiments verify our theoretical results.
Paper Structure (14 sections, 10 theorems, 82 equations, 2 figures)

This paper contains 14 sections, 10 theorems, 82 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Assumptions on SODEs and Useful Facts from Markov Chain Theory
  3. Main Assumptions on SODEs
  4. Preliminaries on Ergodicity of Markov Chains
  5. Geometric Ergodicity of STM for Monotone SODEs
  6. Lyapunov structure
  7. Minorization condition and unique ergodicity of STM
  8. Unique Ergodicity of Galerkin-based Full Discretizations for Monotone SPDEs
  9. Preliminaries and Galerkin-based Full Discretizations
  10. Lyapunov structure of DIEG
  11. Numerical Experiments
  12. Experiments on STM
  13. Experiments on DIEG
  14. Discussions

Key Result

Lemma 1

Let Assumption A1 hold. Then STM scheme stm applied to Eq. sde can be uniquely solved when $L_1 \theta \tau<2$. Moreover, $(X_n)_{n\in \mathbb N}$ is adapted to $(\mathscr F_n)_{n\in \mathbb N}$ and enjoys homogenous Markov property.

Figures (2)

  • Figure 1: Empirical density of \ref{['stm']} for \ref{['sde']}
  • Figure 2: Time averages of \ref{['die-g']} for \ref{['ac']}

Theorems & Definitions (28)

  • Lemma 1
  • proof
  • Example 1
  • Remark 1
  • Lemma 2
  • proof
  • Theorem 1
  • proof
  • Remark 2
  • Corollary 1
  • ...and 18 more