Approximation of invariant measures for random lattice reversible Selkov systems

TL;DR

The paper addresses approximating invariant measures for random lattice reversible Selkov systems on $\mathbb{Z}$ by developing a backward Euler-Maruyama time discretization in $X=\ell^2\times\ell^2$ and constructing numerical invariant measures via Krylov–Bogolyubov. It also analyzes finite-dimensional truncations to enable practical computation and proves upper semicontinuous convergence of the numerical invariant measures to those of the continuous system as $\Delta\to0$. The main contributions are the existence of numerical invariant measures for the BEM scheme, their convergence to the continuous invariant measures, and the validation of finite-dimensional approximations through two-stage limits $N\to\infty$ and $\Delta\to0$. This provides a rigorous framework for approximating stationary distributions of stochastic lattice Selkov dynamics with implications for numerical analysis of random lattice systems.

Abstract

This paper focuses on the numerical approximation of random lattice reversible Selkov systems. It establishes the existence of numerical invariant measures for random models with nonlinear noise, using the backward Euler-Maruyama (BEM) scheme for time discretization. The study examines both infinite dimensional discrete random models and their corresponding finite dimensional truncations. A classical path convergence technique is employed to demonstrate the convergence of the invariant measures of the BEM scheme to those of the random lattice reversible Selkov systems. As the discrete time step size approaches zero, the invariant measure of the random lattice reversible Selkov systems can be approximated by the numerical invariant measure of the finite dimensional truncated systems.

Theorems & Definitions (24)

• Lemma 2.1
• proof
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Lemma 3.3
• proof
• Theorem 3.1
• proof