Pullback measure attractors and evolution system of measures for  <i>p</i> -Laplacian Ginzburg–Landau lattice systems driven by superlinear Lévy noise

Abstract This paper investigates the existence, uniqueness, and asymptotic behavior of pullback measure attractors and evolution systems of measures for a complex-valued p -Laplacian Ginzburg–Landau lattice system ( GLLS ) driven by superlinear Lévy noise. We first derive several long-time uniform estimates and tail-end estimates of the solutions, and establish the existence and uniqueness of pullback measure attractors for the non-autonomous dynamical system generated by the solution operators in the space of probability measures. We then study the existence of evolution systems of measures and analyze their limiting behavior. The upper semicontinuity of the pullback measure attractors is also discussed. Under a large damping assumption, we prove that the pullback measure attractor reduces to a singleton, which yields the existence, uniqueness and exponential mixing of periodic/invariant measures and evolution systems of measures for the corresponding stochastic systems. A key result shows that the limit of a sequence of evolution systems of measures of the p -Laplacian GLLS with superlinear Lévy noise must be an evolution system of measures of the corresponding limiting system. The difficulties caused by the superlinear noise coefficients, the nonlinear p -Laplace operator, the lack of compactness in infinite-dimensional lattices, and the inapplicability of Fatou’s lemma in the non-autonomous case are overcome by using the polynomial dissipation of the drift term, the uniform tail-end estimates, the large-time pullback argument, and the monotonicity-balance conditions. Finally, some numerical simulations are presented to illustrate the behavior of the stochastic GLLS driven by superlinear Lévy noise.