Showing papers by "Nicolas Fournier published in 2014"

Propagation of chaos for the 2D viscous vortex model

Abstract: We consider a stochastic system of $N$ particles, usually called vortices in that setting, approximating the 2D Navier-Stokes equation written in vorticity. Assuming that the initial distribution of the position and circulation of the vortices has finite (partial) entropy and a finite moment of positive order, we show that the empirical measure of the particle system converges in law to the unique (under suitable a priori estimates) solution of the 2D Navier-Stokes equation. We actually prove a slightly stronger result : the propagation of chaos of the stochastic paths towards the solution of the expected nonlinear stochastic differential equation. Moreover, the convergence holds in a strong sense, usually called entropic (there is no loss of entropy in the limit). The result holds without restriction (but positivity) on the viscosity parameter. The main difficulty is the presence of the singular Biot-Savart kernel in the equation. To overcome this problem, we use the dissipation of entropy which provides some (uniform in $N$) bound on the Fisher information of the particle system, and then use extensively that bound together with classical and new properties of the Fisher information.

On a toy model of interacting neurons

Abstract: We continue the study of a stochastic system of interacting neurons introduced in De Masi-Galves-L\"ocherbach-Presutti (2014). The system consists of N neurons, each spiking randomly with rate depending on its membrane potential. At its spiking time, the neuron potential is reset to 0 and all other neurons receive an additional amount 1/N of potential. Moreover, electrical synapses induce a deterministic drift of the system towards its center of mass. We prove propagation of chaos of the system, as N tends to infinity, to a limit nonlinear jumping stochastic differential equation. We consequently improve on the results of De Masi-Galves-L\"ocherbach-Presutti (2014), since (i) we remove the compact support condition on the initial datum, (ii) we get a rate of convergence in $1/\sqrt N$. Finally, we study the limit equation: we describe the shape of its time-marginals, we prove the existence of a unique non-trivial invariant distribution, we show that the trivial invariant distribution is not attractive, and in a special case, we establish the convergence to equilibrium.

High dimensional Hawkes processes

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 {\it single impulsion} started at a given point of $\zz^d$ 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.

A Mean-field forest-fire model

Abstract: We consider a family of discrete coagulation-fragmentation equations closely related to the one-dimensional forest-fire model of statistical mechanics: each pair of particles with masses i, j ∈ N merge together at rate 2 to produce a single particle with mass i + j, and each particle with mass i breaks into i particles with mass 1 at rate (i − 1)/n. The (large) parameter n controls the rate of ignition and there is also an acceleration factor (depending on the total number of particles) in front of the coagulation term. We prove that for each n ∈ N, such a model has a unique equilibrium state and study in details the asymptotics of this equilibrium as n → ∞: (I) the distribution of the mass of a typical particle goes to the law of the number of leaves of a critical binary Galton-Watson tree, (II) the distribution of the mass of a typical size-biased particle converges, after rescaling, to a limit profile, which we write explicitly in terms of the zeroes of the Airy function and its derivative. We also indicate how to simulate perfectly a typical particle and a size-biased typical particle by pruning some random trees.