We consider stochastic evolution equations (SEEs) of parabolic type in Hilbert space with smooth coefficients, driven by multiplicative, not necessarily trace class, Gaussian noise. We present the notion of extended transition semigroups for such equations and we show, under suitable assumptions, that the extended transition semigroup is a solution to the Kolmogorov equation in infinite dimensions. In addition, Fr\'{e}chet differentiability of the extended transition semigroup in negative order spaces is established. The order of smoothness is the same as the order of smoothes of the coefficients of the corresponding equation. In order to define the extended transition semigroup, stochastic evolution equations with irregular initial values and initial singularities in the coefficients are investigated and an abstract existence and uniqueness result for such equations is presented.

中文翻译：

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具有不规则初始值的随机演化方程的存在性、唯一性和规律性

我们考虑 Hilbert 空间中具有平滑系数的抛物线型随机演化方程 (SEE)，由乘法（不一定是迹类）高斯噪声驱动。我们提出了此类方程的扩展转移半群的概念，并证明在适当的假设下，扩展转移半群是无限维 Kolmogorov 方程的解。此外，还建立了负序空间中扩展跃迁半群的Fr\'{e}chet可微性。平滑的阶数与对应方程的系数的平滑阶数相同。为了定义扩展跃迁半群，