## The Annals of Applied Probability

### Hawkes processes on large networks

#### Abstract

We generalise the construction of multivariate Hawkes processes to a possibly infinite network of counting processes on a directed graph $\mathbb{G}$. The process is constructed as the solution to a system of Poisson driven stochastic differential equations, for which we prove pathwise existence and uniqueness under some reasonable conditions.

We next investigate how to approximate a standard $N$-dimensional Hawkes process by a simple inhomogeneous Poisson process in the mean-field framework where each pair of individuals interact in the same way, in the limit $N\rightarrow\infty$. In the so-called linear case for the interaction, we further investigate the large time behaviour of the process. We study in particular the stability of the central limit theorem when exchanging the limits $N,T\rightarrow\infty$ and exhibit different possible behaviours.

We finally consider the case $\mathbb{G}=\mathbb{Z}^{d}$ with nearest neighbour interactions. In the linear case, we prove some (large time) laws of large numbers and exhibit different behaviours, reminiscent of the infinite setting. Finally, we study the propagation of a single impulsion started at a given point of $\mathbb{Z}^{d}$ at time $0$. We compute the probability of extinction of such an impulsion and, in some particular cases, we can accurately describe how it propagates to the whole space.

#### Article information

Source
Ann. Appl. Probab., Volume 26, Number 1 (2016), 216-261.

Dates
Revised: November 2014
First available in Project Euclid: 5 January 2016

https://projecteuclid.org/euclid.aoap/1452003238

Digital Object Identifier
doi:10.1214/14-AAP1089

Mathematical Reviews number (MathSciNet)
MR3449317

Zentralblatt MATH identifier
1334.60082

#### Citation

Delattre, Sylvain; Fournier, Nicolas; Hoffmann, Marc. Hawkes processes on large networks. Ann. Appl. Probab. 26 (2016), no. 1, 216--261. doi:10.1214/14-AAP1089. https://projecteuclid.org/euclid.aoap/1452003238

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