Random Pullback Attractors and Numerical Approximation for Nonlocal Lattice Delay Equations Driven by Rough Noise
Abstract
We investigate the pathwise asymptotic dynamics of a nonlocal lattice equation with fixed delay on ℓ2(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.
Related
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BibTeX
@misc{zhang2026random,
title = {{Random Pullback Attractors and Numerical Approximation for Nonlocal Lattice Delay Equations Driven by Rough Noise}},
author = {Lijuan Zhang and Jiangwei Zhang and Jianhua Huang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.07076},
eprint = {2610.07076},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.07076},
}
RIS
TY - UNPB
TI - Random Pullback Attractors and Numerical Approximation for Nonlocal Lattice Delay Equations Driven by Rough Noise
AU - Lijuan Zhang
AU - Jiangwei Zhang
AU - Jianhua Huang
PY - 2026
DA - 2026/10/05
PB - arXiv
DO - 10.48550/arxiv.2610.07076
UR - https://arxiv.org/abs/2610.07076
AB - We investigate the pathwise asymptotic dynamics of a nonlocal lattice equation with fixed delay on ℓ^2(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.
ER -