Abstract
The investigation into the dynamic behavior of infinite lattice systems holds paramount significance in the realm of physical phenomena, particularly in mechanics. This intricate domain has captivated the attention of both mathematicians and physicists.In acknowledgment of the inherent noise prevalent in real-world environments, our study embraces this aspect by introducing a random term into our model. This deliberate inclusion of stochasticity engenders a novel perspective, giving rise to a stochastic lattice differential equation. This model proves to be a versatile tool for accurately characterizing spatial structures characterized by discrete components and the associated uncertainties that pervade them. This research elucidates the intricate interplay between lattice dynamics and environmental noise, shedding light on the complex behavior of such systems in a realistic context. Our result generalizes many results in three directions: extending the connections between the terms to non-linear, extending the connection neighborhood from 3 (as in most cases) to arbitrary value n, and extending the results that are in ℓ2 to ℓρ2.
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