Scaling Limits of Solutions of Linear Evolution Equations with Random Initial Conditions

We consider a linear equation $\partial_t u = \mathcal{L}u$, where $\mathcal{L}$ is a generator of a semigroup of linear operators on a certain Hilbert space related to an initial condition $u(0)$ being a generalised stationary random field on $\mathbb{R}^d$. We show the existence and uniqueness of generalised solutions to such initial value problems. Then we investigate their scaling limits.