Fluctuations for Spatially Extended Hawkes Processes

Julien Chevallier; Guilherme Ost

Home/Archive/arXiv

arXiv · 1909.06401

Fluctuations for Spatially Extended Hawkes Processes

Julien Chevallier·Guilherme Ost

View original source ↗Read source document ↗Export metadata ↓

Abstract

In a previous paper, it has been shown that the mean-field limit of spatially extended Hawkes processes is characterized as the unique solution $u(t,x)$ of a neural field equation (NFE). The value $u(t,x)$ represents the membrane potential at time $t$ of a typical neuron located in position $x$, embedded in an infinite network of neurons. In the present paper, we complement this result by studying the fluctuations of such a stochastic system around its mean-field limit $u(t,x)$. Our first main result is a central limit theorem stating that the spatial distribution associated with these fluctuations converges to the unique solution of some stochastic differential equation driven by a Gaussian noise. In our second main result, we show that the solutions of this stochastic differential equation can be well approximated by a stochastic version of the neural field equation satisfied by $u(t,x)$. To the best of our knowledge, this result appears to be new in the literature.

Explore related subjects

math.PR

Keep this discovery

BibTeX RIS

Julien Chevallier, Guilherme Ost. 2019-09-13. Fluctuations for Spatially Extended Hawkes Processes. https://arxiv.org/abs/1909.06401

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Averaging principles for nonautonomous multiscale stochastic Burgers equations with reflection

In this paper, we study averaging principles for nonautonomous multiscale stochastic Burgers equations with reflection. First, we derive a general averaging principle applicable to such equations under minimal assumptions. Subsequently, since the coefficients of the obtained averaged equation still depend on the small scaling parameter $\e$, we impose either periodic or asymptotic conditions on the coefficients, thereby obtain two distinct averaged equations whose coefficients are independent of $\e$ and establish two averaging principles. Stopping times and Khasminskii's time discretization schemes play an important role. Finally, a concrete example is provided to illustrate the applicability and validity of the theoretical results.

math.PR↗

arXiv2026

Spectral properties of Random Matrices

We give the theoretical foundations of random matrix theory through the definitions of a random matrix, a random probability measure and the corresponding empirical spectral distribution. The technical tool we use is the Stieltjes transform method through which we prove optimal convergence of the empirical spectral distribution of random sample covariance matrices to the deterministic Marchenko-Pastur distribution. We also give new results about the rigidity of the eigenvalues of this random sample covariance matrix and the rate of their convergence. We then define the Dyson equation method to prove new local laws about a random matrix model that interpolates between the Marchenko-Pastur distribution, the elliptical law and the circular law. Through our work these local laws can be considered universal.

math.PR↗

arXiv2026

Moments approach for the elephant random walk

We discuss the method of moments for the one-dimensional elephant random walk (ERW). We first derive a differential recurrence relation for the characteristic function of the ERW, which yields a corresponding system of recurrence relations for its moments. We then obtain asymptotic approximations for the moments in each of the three parameter regimes of the ERW. Finally, by establishing the convergence of the moments and verifying the corresponding moment-determinacy conditions, we identify the limiting distributions of the ERW in each regime.

math.PR↗