Showing papers in "ALEA-Latin American Journal of Probability and Mathematical Statistics in 2007"

Hitting probabilities for systems of non-linear stochastic heat equations with additive noise

Abstract: We consider a system of d coupled non-linear stochastic heat equations in spatial dimension 1 driven by d-dimensional additive space-time white noise. We establish upper and lower bounds on hitting probabilities of the solution {u(t , x)}t∈R+,x∈[0 ,1], in terms of respectively Hausdorff measure and Newtonian capacity. We also obtain the Hausdorff dimensions of level sets and their projections. A result of independent interest is an anisotropic form of the Kolmogorov continuity theorem. AMS 2000 subject classifications: Primary: 60H15, 60J45; Secondary: 60G60.

The Stochastic Heat Equation with a Fractional-Colored Noise: Existence of the Solution

Abstract: In this article we consider the stochastic heat equation ut −�u = u B in (0,T) ×R d , with vanishing initial conditions, driven by a Gaussian noise u B which is fractional in time, with Hurst index H ∈ (1/2,1), and colored in space, with spatial covariance given by a function f. Our main result gives the necessary and sufficient condition on H for the existence of a solution. When f is the Riesz kernel of order � ∈ (0,d) this condition is H > (d −�)/4, which is a relaxation of the condition H > d/4 encountered when the noise u B is white in space. When f is the Bessel

Shape and local growth for multidimensional branching random walks in random environment

Abstract: We study branching random walks in random environment on the d- dimensional square lattice, d 1. In this model, the environment has nite range dependence, and the population size cannot decrease. We prove limit theorems (laws of large numbers) for the set of lattice sites which are visited up to a large time as well as for the local size of the population. The limiting shape of this set is compact and convex, and the local size is given by a concave growth exponent. Also, we obtain the law of large numbers for the logarithm of the total number of particles in the process. 1. Introduction and results We start with an informal description of the model we study in this paper. Particles live in Z d and evolve in discrete time. At each time, every particle is substituted by (possibly more than one) ospring which are placed in neighboring sites, independently of the other particles. The rules of ospring generation depend only on the location of the particle. The collection of those rules (so-called the environment) is itself random, it is chosen randomly before starting the process, and then it is kept xed during all the subsequent evolution of the particle system.

On Stein's method and perturbations

Abstract: Stein's (1972) method is a very general tool for assessing the quality of approximation of the distribution of a random element by another, often sim- pler, distribution. In applications of Stein's method, one needs to establish a Stein identity for the approximating distribution, solve the Stein equation and estimate the behaviour of the solutions in terms of the metrics under study. For some Stein equations, solutions with good properties are known; for others, this is not the case. Barbour and Xia (1999) introduced a perturbation method for Poisson approxima- tion, in which Stein identities for a large class of compound Poisson and translated Poisson distributions are viewed as perturbations of a Poisson distribution. In this paper, it is shown that the method can be extended to very general settings, including perturbations of normal, Poisson, compound Poisson, binomial and Pois- son process approximations in terms of various metrics such as the Kolmogorov, Wasserstein and total variation metrics. Examples are provided to illustrate how the general perturbation method can be applied.

Weakly dependent random fields with infinite interactions - paru sous le titre "A fixed point approach to model random fields"

Abstract: We introduce new models of stationary random fields, solutions of Xt = F ( (Xt−j)j∈Zd\{0}; ξt ) , the input random field ξ is stationary, e.g. ξ is independent and identically distributed (iid). Such models extend most of those used in statistics. The (nontrivial) existence of such models is based on a contraction principle and Lipschitz conditions are needed; those assumptions imply Doukhan and Louhichi (1999)’s [6] weak dependence conditions. In contrast to the concurrent ones, our models are not set in terms of conditional distributions. Various examples of such random fields are considered. We also use a very weak notion of causality of independent interest: it allows to relax the boundedness assumption of inputs for several new heteroscedastic models, solutions of a nonlinear equation. Primary AMS keyword: 60G60: Random fields keywords: 60B12: Limit theorems for vector-valued random variables, 60F25: L-limit theorems, 60K35: Interacting random processes; statistical mechanics type models; percolation theory, 62M40: Random fields; image analysis, 60B99: Weak dependence, 60K99: Bernoulli shifts.

Hurst exponent estimation of Fractional Lévy Motion

Abstract: In this paper, we build an estimator of the Hurst exponent of a frac- tional L evy motion. The stochastic process is observed with random noise errors in the following framework: continuous time and discrete observation times. In both cases, we prove consistency of our wavelet type estimator. Moreover we perform some simulations in order to study numerically the asymptotic behaviour of this estimate.

On the genealogy of conditioned stable Lévy forests

Abstract: A Levy forest of size s > 0 is a Poisson point process in the set of Levy trees which is defined on the time interval (0, s). The total mass of this forest is defined by the sum of the masses of all its trees. We give a realization of the stable Levy forest of a given size conditioned on its total mass using the path of the uncon- ditioned forest. Then, we prove an invariance principle for this conditioned forest by considering k independent Galton-Watson trees whose offspring distribution is in the domain of attraction of any stable law conditioned on their total progeny to be equal to n. We prove that when n and k tend towards +∞, under suitable rescaling, the associated coding random walk, the contour and height processes all converge in law on the Skorokhod space towards the first passage bridge and height process of a stable Levy process with no negative jumps respectively.