# Self-similar behaviour of a non-local diffusion equation with time delay

## TL;DR

This paper investigates the long-term behavior of solutions to a class of non-local measure-valued differential equations with time delay, showing they behave like heat kernels asymptotically, with implications for stochastic processes with memory.

## Contribution

It introduces a novel analysis of the asymptotic behavior of delayed non-local equations using self-similar rescaling and Young measures, extending understanding of their long-term dynamics.

## Key findings

- Solutions exhibit heat kernel-like behavior asymptotically
- Convergence established via Young measures and fractional Sobolev estimates
- Results apply to stochastic processes with memory

## Abstract

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.