Random dynamics of stochastic retarded parabolic equations with multiplicative noise on ℝN

In this paper, we study the long-term behavior and its regularity for a class of non-autonomous retarded stochastic integro-differential parabolic equations influenced by multiplicative white noise on the [Formula: see text]-dimensional Euclidean space. A non-random hypothesis thoroughly depending on the deterministic parameters of the problem is presented to guarantee the existence of a tempered pullback random attractor in the intersection space of [Formula: see text] and [Formula: see text], where [Formula: see text] signifies the dissipation exponent of a general nonlinear forcing. We circumvent an impediment of the non-compactness of solutions on unbounded domains to verify the asymptotical compactness in [Formula: see text] by utilizing the uniform tail estimates along with the Arzela–Ascoli theorem. The asymptotical compactness in [Formula: see text] is proved jointly by [Formula: see text]-asymptotical compactness and a new truncation estimate technique.