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.
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机译：我们考虑一类由乘性高斯噪声驱动的半线性Volterra型随机演化方程。内存内核（不一定是解析的）使得确定性线性方程式呈现抛物线形。在适当的非线性项的Lipschitz型和线性增长假设下，我们证明了唯一的温和解是均值-$ p $ H \“ oldercontinuous，具有适当的Sobolev空间中的值，具体取决于内核和数据。尤其是，我们获得了路径空间解的时间（Sobolev-H“较旧”）正则性以及空间Sobolev范数上的最大类型。作为主要技术工具之一，我们建立了确定性进化算子族的导数的平滑属性。
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