System Of Independent Random Walks Research Articles

Large deviations in the supremum norm for a reaction–diffusion system

We present large deviations estimates in the supremum norm for a system of independent random walks superposed with a birth-and-death dynamics evolving on the discrete torus with N sites. The scaling limit considered is the so-called high density limit (see the survey in (In From Particle Systems to Partial Differential Equations (2014) 179–189 Springer) on the subject), where space, time and initial quantity of particles are rescaled. The associated rate functional here obtained is a semi-linearized version of the rate function of (Probability Theory and Related Fields 97 (1992) 339–361), which dealt with large deviations of exclusion processes superposed with birth-and-death dynamics. An important ingredient in the proof of large deviations consists in providing a limit of a suitable class of perturbations of the original process, which is precisely one of the main contributions of this work: a strategy to extend the original high density approach (as in Advances in Applied Probability 12 (1980) 367–379; Annals of Probability 19 (1991) 1440–1462; The Annals of Applied Probability 2 (1992) 131–141; Journal of Statistical Physics 149 (2012) 629–642; Annals of Probability 14 (1986) 173–193; Probability Theory and Related Fields 78 (1988) 11–37) to weakly asymmetric systems. Two cases are considered with respect to the initial quantity of particles, the power law and the (at least) exponential growth. In the first case, we present the lower bound only on a certain set of smooth profiles, while in the second case under some extra technical assumptions we provide a full large deviations principle.

Scaling limit of a directed polymer among a Poisson field of independent walks

We consider a directed polymer model in dimension 1+1, where the disorder is given by the occupation field of a Poisson system of independent random walks on Z. In a suitable continuum and weak disorder limit, we show that the family of quenched partition functions of the directed polymer converges to the Stratonovich solution of a multiplicative stochastic heat equation (SHE) with a Gaussian noise, whose space-time covariance is given by the heat kernel.In contrast to the case with space-time white noise where the solution of the SHE admits a Wiener-Itô chaos expansion, we establish an L1-convergent chaos expansions of iterated integrals generated by Picard iterations. Using this expansion and its discrete counterpart for the polymer partition functions, the convergence of the terms in the expansion is proved via functional analytic arguments and heat kernel estimates. The Poisson random walk system is amenable to careful moment analysis, which is an important input to our arguments.

Hydrodynamic limit for a system of independent, sub-ballistic random walks in a common random environment

Nous considérons un système de particules indépendantes évoluant dans un milieu aléatoire commun. Auparavant, la limite hydrodynamique de ce système de particules en milieu aléatoire a été obtenue quand les particules sont transientes avec une vitesse positive (Electron. J. Probab. 15 (2010) 1024–1040). Dans cet article nous considérons le cas où les particules sont transientes mais ont une vitesse sous-linéaire d’ordre $n^{\kappa}$ pour $\kappa\in(0,1)$ et nous montrons l’existence d’une limite hydrodynamique du système de particules avec une échelle du temps $n^{1/\kappa}$ et une échelle spatiale $n$. La propriété la plus intéressante de cette limite hydrodynamique est que le milieu n’est pas moyenné par la limite ; c’est-à-dire, la densité asymptotique des particules dépend de la réalisation du milieu choisi. La limite hydrodynamique du système de particules est déduite à partir de la limite hydrodynamique d’un système de particules indépendantes dans un milieu aléatoire dirigé.

A Particle System with Explosions: Law of Large Numbers for the Density of Particles and the Blow-Up Time

Consider a system of independent random walks in the discrete torus with creation-annihilation of particles and possible explosion of the total number of particles in finite time. Rescaling space and rates for diffusion/creation/annihilation of particles, we obtain a stong law of large numbers for the density of particles in the supremum norm. The limiting object is a classical solution to the semilinear heat equation u_t =u_{xx} + f(u). If f(u)=u^p, 1<p \le 3, we also obtain a law of large numbers for the explosion time.

Quasi-Stationary measures for conservative dynamics in the infinite lattice

We study quasi-stationary measures for conservative particle systems in the in finite lattice. Existence of quasi-stationary measures is established for a fairly general class of reversible systems. For the special cases of a system of independent random walks and the symmetric simple exclusion process, it is shown that qualitative features of quasi-stationary measures change drastically with dimension.

First occurrence time of a large density fluctuation for a system of independent random walks

We obtain sharp asymptotics for the first time a “macroscopic” density fluctuation occurs in a system of independent simple symmetric random walks on Z d . Also, we show the convergence of the moments of the rescaled time by establishing tail estimates.

Large deviations for independent random walks

We consider a system of independent random walks on ℤ. Let ξ n (x) be the number of particles atx at timen, and letL n (x)=ξ0(x)+ ... +ξ n (x) be the total occupation time ofx by timen. In this paper we study the large deviations ofL n (0)−L n (1). The behavior we find is much different from that ofL n (0). We investigate the limiting behavior when the initial configurations has asymptotic density 1 and when ξ0(x) are i.i.d Poisson mean 1, finding that the asymptotics are different in these two cases.