Scaling limits of solutions of linear evolution equations with random initial conditions

We consider a linear equation ∂tu = ℒu, where 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 ℝd. We show the existence and uniqueness of generalised solutions to such initial value problems. Then we investigate their scaling limits.