Random Pullback Attractors and Numerical Approximation for Nonlocal Lattice Delay Equations Driven by Rough Noise

We investigate the pathwise asymptotic dynamics of a nonlocal lattice equation with fixed delay on ${\mathrm{\ell }}^2(\mathbb{Z})$ 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.