Weak and generalized with random variable solutions of stochastic Сauchy 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 $\mathrm{H}$ and a linear operator $\mathrm{B: H}\to H$. 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.