Spatiotemporal Hawkes processes with a graphon-induced connectivity structure

par Justin Baars

mardi 25 mars 2025, 09:50 → 10:50 Europe/Paris

We introduce a spatiotemporal self-exciting point process $(N_t(x))$, boundedly finite both over time $[0,\mathrm{\infty })$ and space $\mathcal{X}$, with excitation structure determined by a graphon $W$ on ${\mathcal{X}}^2$. This graphon Hawkes process generalizes both the multivariate Hawkes process and the Hawkes process on a countable network, and despite being infinite-dimensional, it is surprisingly tractable. After proving existence, uniqueness and stability results, we show, both in the annealed and in the quenched case, that for compact, Euclidean $\mathcal{X}\subset {\mathbb{R}}^m$, any graphon Hawkes process can be obtained as the suitable limit of $d$-dimensional Hawkes processes ${\tilde{N}}^d$, as $d\to \mathrm{\infty }$. Furthermore, in the stable regime, we establish an FLLN and an FCLT for our infinite-dimensional process on compact $\mathcal{X}\subset {\mathbb{R}}^m$, while in the unstable regime we prove divergence of $N_T(\mathcal{X})/T$, as $T\to \mathrm{\infty }$. Finally, we exploit a cluster representation to derive fixed-point equations for the Laplace functional of $N$, for which we set up a recursive approximation procedure. We apply these results to show that, starting with multivariate Hawkes processes ${\tilde{N}}_t^d$ converging to stable graphon Hawkes processes, the limits $d\to \mathrm{\infty }$ and $t\to \mathrm{\infty }$ commute.