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We consider a class of semilinear Volterra type stochastic evolution equationdriven by multiplicative Gaussian noise. The memory kernel, not necessarilyanalytic, is such that the deterministic linear equation exhibits a paraboliccharacter. Under appropriate Lipschitz-type and linear growth assumptions onthe nonlinear terms we show that the unique mild solution is mean-$p$ H\"oldercontinuous with values in an appropriate Sobolev space depending on the kerneland the data. In particular, we obtain pathwise space-time (Sobolev-H\"older)regularity of the solution together with a maximal type bound on the spatialSobolev norm. As one of the main technical tools we establish a smoothingproperty of the derivative of the deterministic evolution operator family.

Existence, uniqueness and regularity for a class of semilinear stochastic Volterra equations with multiplicative noise