Mean-field limit of generalized Hawkes processes

We generalize multivariate Hawkes processes mainly by including a dependence with respect to the age of the process, i.e. the delay since the last point.Within this class, we investigate the limit behaviour, when $n$ goes to infinity, of a system of $n$ mean-field interacting age-dependent Hawkes processes. We prove that such a system can be approximated by independent and identically distributed age dependent point processes interacting with their own mean intensity. This result generalizes the study performed by Delattre et al. (2016).In continuity with Chevallier et al. (2015), the second goal of this paper is to give a proper link between these generalized Hawkes processes as microscopic models of individual neurons and the age-structured system of partial differential equations introduced by Pakdaman et al. (2010) as macroscopic model of neurons.     

Numerical solution of a Cauchy problem for nonlinear reaction diffusion processes

In this paper, the authors propose a numerical method to compute the solution of the Cauchy problem: _wt-(w^m_wx)x=w^p$w_t-(w^m{w}_x{)}_x=w^p$, the initial condition is a nonnegative function with compact support, m>0$m>0$, p?m+1$p?m+1$. The problem is split into two parts: a hyperbolic term solved by using the Hopf and Lax formula and a parabolic term solved by a backward linearized Euler method in time and a finite element method in space. The convergence of the scheme is obtained. Further, it is proved that if m+1?p<m+3$m+1?p<m+3$, any numerical solution blows up in a finite time as the exact solution, while for p>m+3$p>m+3$, if the initial condition is sufficiently small, a global numerical solution exists, and if p?m+3$p?m+3$, for large initial condition, the solution is unbounded.     

On large deviations for particle systems associated with spatially homogeneous Boltzmann type equations

Summary We consider a dynamical interacting particle system whose empirical distribution tends to the solution of a spatially homogeneous Boltzmann type equation, as the number of particles tends to infinity. These laws of large numbers were proved for the Maxwellian molecules by H. Tanaka [Tal] and for the hard spheres by A.S. Sznitman [Szl]. In the present paper we investigate the corresponding large deviations: the large deviation upper bound is obtained and, using convex analysis, a non-variational formulation of the rate function is given. Our results hold for Maxwellian molecules with a cutoff potential and for hard spheres.     

State estimation for cox processes with unknown probability law

Let N_i, i?1, be i.i.d. observable Cox processes on a compact metric space E, directed by unobservable random measures M_i. Assume that the probability law of the M_i is completely unknown. Techniques are developed for approximation of state estimators using data from the processes N₁,…,N_n to estimate necessary attributes of the unknown probability law of the time M_i. The techniques are based on representation of the state estimators in terms of reduced Palm distributions of the N_i and on estimation of these Palm distributions. Estimators of Palm distributions are shown to be strongly consistent and asymptotically normal. The difference between the true and the pseudo-state estimators converges to zero in L² at rate n^{?$\text{1}\text{4+δ}$} for each δ > 0.     

Partial mean field limits in heterogeneous networks

We investigate systems of interacting stochastic differential equations with two kinds of heterogeneity: one originating from different weights of the linkages, and one concerning their asymptotic relevance when the system becomes large. To capture these effects, we define a partial mean field system, and prove a law of large numbers with explicit bounds on the mean squared error. Furthermore, a large deviation result is established under reasonable assumptions. The theory will be illustrated by several examples: on the one hand, we recover the classical results of chaos propagation for homogeneous systems, and on the other hand, we demonstrate the validity of our assumptions for quite general heterogeneous networks including those arising from preferential attachment random graph models.     

Heat Equations with Fractional White Noise Potentials

This paper is concerned with the following stochastic heat equations: where w ^H is a time independent fractional white noise with Hurst parameter H=(h ₁ , h ₂ ,..., h _d ) , or a time dependent fractional white noise with Hurst parameter H=(h ₀ , h ₁ ,..., h _d ) . Denote |H|=h ₁ +h ₂ +...+h _d . When the noise is time independent, it is shown that if ? <h _i <1 for i=1, 2,..., d and if |H|>d-1 , then the solution is in L ₂ and the L ₂ -Lyapunov exponent of the solution is estimated. When the noise is time dependent, it is shown that if ? <h _i <1 for i=0, 1,..., d and if |H|>d- 2 /( 2h ₀ -1 ) , the solution is in L ₂ and the L ₂ -Lyapunov exponent of the solution is also estimated. A family of distribution spaces S _ρ , ρ∈ RR , is introduced so that every chaos of an element in S _ρ is in L ₂ . The Lyapunov exponents in S _ρ of the solution are also estimated. Accepted 10 October 2000. Online publication 19 February 2001.