Connection between weak and generalized solutions of infinite-dimensional stochastic problems

We investigate the stochastic Cauchy problem for the first order equation with singular white noise and generators of regularized (integrated, convoluted) semigroups in Hilbert spaces and abstract distribution spaces. Weak solutions for the problem in the Ito form and generalized solutions for the differential problem in abstract distribution spaces are constructed in dependence on properties of the generator. Connections between these solutions are shown.