Gaussian white noise with trajectories in the space $\mathcal S'(H)$

In this paper we construct a Gaussian white noise with trajectories in the space of generalized functions over $\mathcal S$ with values in a separable Hilbert space $H$. We obtain a solution to the Cauchy problem for a linear operator-differential equation with the additive white noise as a generalized random process with trajectories in the space of exponential distributions. We prove existence of the solution in the case when the operator coefficient $A$ generates a $C_0$ semigroup and in the case when $A$ generates an integrated semigroup.