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We consider a class of infinite delay equations in Banach spaces that models arising in the theory of viscoelasticity, for instance. The equation involves a completely monotone convolution kernel with a singularity at t = 0 and a sectorial linear spatial operator. Our main goal here is the construction of a semigroup formulation for the integral equation; in the last part of the paper, we illustrate the potentiality of the approach by considering a stochastic perturbation of the problem. Existence and uniqueness of a weak solution is established. The corresponding evolutionary solution process is Markovian, and the tools of linear analytic semigroup theory can be utilized.

Volterra equations in Banach spaces with completely monotone kernels