Let and be real separable Hilbert spaces. Assume that is a -valued Wiener process with covariance operator and is an -valued random variable which is independent of . Consider the initial value problem of semilinear functional integro-differential stochastic evolution equations
with values in , where represents a linear operator, , with and . The authors discuss global existence results concerning mild and periodic solutions under several growth and compactness conditions. Weak convergence of induced probability measures belonging to the family of finite-dimensional distributions of certain sequences of such stochastic equations is treated too. Basic proof-tools include Schaefer’s fixed point theorem, techniques of linear semigroups and probability measures as well as results from infinite-dimensional SDEs. Conceivable applications to electromagnetic theory, population dynamics and heat conduction in materials with memory underline the importance of their work. An example of a nonlocal integro-partial SDE illustrates some thoughts of related abstract theory. Some necessary preliminaries compiled from probability theory and functional analysis ease the process of understanding by lesser experienced readership.