Existence, uniqueness and regularity for stochastic evolution equations with irregular initial values

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.