Regularity of pullback random attractors and invariant sample measures for nonautonomous stochastic <inline-formula><tex-math id="M1">$ p $</tex-math></inline-formula>-Laplacian lattice systems

We consider a nonautonomous stochastic $ p $-Laplacian lattice equation with multiplicative noise and a nonlinearity that is not locally Lipschitz. For each $ q\geq1 $, a pullback $ (\ell^2, \ell^q) $-attractor is obtained, and the measurability of the pullback attractor in both spaces by more complicated estimates. Then, the time-dependent invariant sample Borel probability measures are constructed and carried by the pullback random attractor. Moreover, the invariant sample measures satisfy a stochastic Liouville type theorem.