Robust dynamics for compact-valued cocycle of stochastic Schrödinger lattice equation with Wong-Zakai noise

We study Wong-Zakai approximations and random attractors of complex stochastic Schrödinger lattice systems driven by non-Lipschitz sources and nonlinear diffusion noise. The non-Lipschitz continuity of both drift and diffusion terms leads to possible multi-solutions and hence the lattice equation generates a multi-valued cocycle, which is proved to be measurable and compact-valued. We first show the existence of a pullback attractor for the approximate system, and then apply the weak upper semicontinuity of multi-valued functions and a countable decomposition of the Wiener probability space to prove the measurability of the pullback attractor as well as the multi-valued cocycle in the original probability space (rather than only in its completion). We finally establish the upper semi-convergence of random attractors from the approximate system to the original Schrödinger lattice system driven by linear multiplicative white noise, as the step size of Wong-Zakai noise tends to zero.