Random Pullback Attractors and Numerical Approximation for Nonlocal Lattice Delay Equations Driven by Rough Noise
Year: 2026
Type: preprint
Abstract: We investigate the pathwise asymptotic dynamics of a nonlocal lattice equation with fixed delay on $\ell^2(\mathbb Z)$ driven by an infinite-dimensional stationary weakly geometric rough path. Using a rough-flow Doss-Sussmann transformation and the method of steps, we establish global well-posedness in the controlled-path framework and construct a continuous random dynamical system. We further construct a tempered pullback absorbing family and use a spatially localized Lyapunov-Krasovskii functional, together with a sewing estimate for the spatial tails of the driving rough path, to prove that the lattice tails vanish uniformly. The existence of a unique random pullback attractor is then established by combining this lattice-tail estimate with an Arzelà--Ascoli compactness argument. Finally, for the delay-compatible compensated Euler schemes and finite-dimensional lattice truncations, we establish uniform absorbing and lattice-tail estimates, which yield the existence of the approximate random pullback attractors and their joint upper semicontinuity, without a coupling condition between the time step and the spatial truncation level.
Source: arXiv (Cornell University)
Authors: Zhang, Lijuan, Zhang, Jiangwei, Huang, Jianhua
Language: English
Subfield: Control and Systems Engineering
Field: Engineering
Domain: Physical Sciences
Open Access status: green
Award: G5567900484
OpenAlex ID: W7220937936