Existence and approximation of measure attractors and invariant measures for McKean-Vlasov stochastic lattice system with Lévy noise

This paper is devoted to the existence and approximation of measure attractors and invariant measures for superlinear McKean-Vlasov stochastic reaction-diffusion lattice system driven by Lévy noise. We firstly prove the well-posedness of solutions by the fixed point arguments. Then by the uniform pullback estimates and tail-ends estimates of solutions, we establish the pullback asymptotic compactness of non-autonomous dynamical systems generated by the solution operators in a space of probability measures, and further obtain the existence and upper semicontinuity of measure attractors. Moreover, we yield the existence and uniqueness of invariant measures as well as ergodicity of the solutions under additional conditions. In addition, the convergence rate of invariant measures is provided when distribution dependent stochastic system converges to distribution independent one. Finally, the finite-dimensional approximations of measure attractors and invariant measures are investigated between such lattice system and its finite-dimensional truncated system, which are useful for studying numerical invariant measures.