Regularity and backward compactness of attractors for non-autonomous lattice systems with random coefficients

We study longtime behavior for the non-autonomous lattice model with multiplicative white noise and a random coefficient in the discrete Laplace operator. We first show existence of a bi-spatial attractor when the initial space is the weighted square summation space and the terminal space is the weighted p-times summation space for p > 2. We then show backward compactness of the attractor in both initial and terminal spaces if the time-indexed forces are backward-tempered and backward-null. Finally, by proving identity of the attractors on the different universes of tempered or backward tempered sets, we show measurability of the attractor in the initial space and in the terminal space, respectively. (C) 2019 Elsevier Inc. All rights reserved.