Weak and generalized with random variable solutions of stochastic Cauchy problem with additive white noise

The article describes the solutions of an abstract stochastic Cauchy problem for the X´(t) = AX(t)+BW(t) equation with the A operator, which is the generator of a semigroup of C 0 class in a Hilbert space H with the white noise W in a different Hilbert space H and a linear operator B: H→H. Two approaches to solve the problem are considered: the Ito integral approach, when the integral problem is solved with ito integral following Wiener process; the approach based on the analysis of the white noise in the original differential problem in the function spaces generalized with random variable. The relation between the solutions is defined.