  We study the asymptotic behaviour of solutions of a class of linear non-local
measure-valued differential equations with time delay. Our main result states
that the solutions asymptotically exhibit a parabolic like behaviour in the
large times, that is precisely expressed in term of heat kernel. Our proof
relies on the study of a-self-similar-rescaled family of solutions. We first
identify the asymptotic behaviour of the solutions by deriving a convergence
result in the sense of the Young measures. Then we strengthen this convergence
by deriving suitable fractional Sobolev compactness estimates. As a by-product,
our main result allows to obtain asymptotic results for a class of piecewise
constant stochastic processes with memory.
