Longtime behavior for the occupation time process of a super-Brownian motion with random immigration

Longtime behavior for the occupation time of a super-Brownian motion with immigration governed by the trajectory of another super-Brownian motion is considered. Central limit theorems are obtained for dimensions d⩾3 that lead to some Gaussian random fields: for 3⩽d⩽5, the field is spatially uniform, which is caused by the randomness of the immigration branching; for d⩾7, the covariance of the limit field is given by the potential operator of the Brownian motion, which is caused by the randomness of the underlying branching; and for d=6, the limit field involves a mixture of the two kinds of fluctuations. Some extensions are made in higher dimensions. An ergodic theorem is proved as well for dimension d=2, which is characterized by an evolution equation.     

On the asymptotic behaviour of the empirical random field of the brownian motion

Let ξ_t, t ? 0, be a d-dimensional Brownian motion. The asymptotic behaviour of the random field ??∫^t₀?(ξ_s) ds is investigated, where ? belongs to a Sobolev space of periodic functions. Particularly a central limit theorem and a law of iterated logarithm are proved leading to a so-called universal law of iterated logarithm.     

Large Deviations for the Super-Brownian Motion with Super-Brownian Immigration

Local large deviation principles are established in dimensions d3 for the super Brownian motion with random immigration X _t , where the immigration rate is governed by the trajectory of another super-Brownian motion . The speed function is t for d4 and t ^1/2 for d=3, compared with the existing results, the interesting phenomenon happened in d=4 with speed t (although only the upper large deviation bound is derived here) is just because the structure of this new model: the random immigration smooth the critical dimension in some sense. The rate function are characterized by an evolution equation.     

Confinement of Brownian motion among Poissonian obstacles in ℝ d , d≥ 3

Consider Brownian motion among random obstacles obtained by translating a fixed compact nonpolar subset of ℝ ^d , d≥ 1, at the points of a Poisson cloud of constant intensity v <: 0. Assume that Brownian motion is absorbed instantaneously upon entering the obstacle set. In SZN-conf Sznitman has shown that in d = 2, conditionally on the event that the process does not enter the obstacle set up to time t, the probability that Brownian motion remains within distance ∼t ^1/4 from its starting point is going to 1 as t goes to infinity. We show that the same result holds true for d≥ 3, with t ^1/4 replaced by t ^{1/( d +2)}. The proof is based on Sznitmans refined method of enlargement of obstacles [10] as well as on a quantitative isoperimetric inequality due to Hall [4]. Received: 6 July 1998     

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We establish a quenched central limit theorem (CLT) for the branching Brownian motion with random immigration in dimension $d\geq4$. The limit is a Gaussian random measure, which is the same as the annealed central limit theorem, but the covariance kernel of the limit is different from that in the annealed sense when d=4.     

Catalytic branching random walks in ℤ d with branching at the origin

A time-continuous branching random walk on the lattice ? ^d , d ≥ 1, is considered when the particles may produce offspring at the origin only. We assume that the underlying Markov random walk is homogeneous and symmetric, the process is initiated at moment t = 0 by a single particle located at the origin, and the average number of offspring produced at the origin is such that the corresponding branching random walk is critical. The asymptotic behavior of the survival probability of such a process at moment t → ∞ and the presence of at least one particle at the origin is studied. In addition, we obtain the asymptotic expansions for the expectation of the number of particles at the origin and prove Yaglom-type conditional limit theorems for the number of particles located at the origin and beyond at moment t.     

Random fields associated with multiple points of the Brownian motion

Let X_t be the Brownian motion in $\text{R}$^d. The random set Γ = {(t₁,…, t_n, z): X_tl = ··· = X_tn = z} in $\text{R}$^{d + n} is empty a.s. except in the following cases: (a) n = 1, d = 1, 2,…; (b) d = 2, n = 2, 3,…; (c) d = 3, n = 2. In each of these cases, a family of random measures M_λ concentrated on Γ is constructed (λ takes values in a certain class of measures on $\text{R}$^d). Measures M_λ characterize the time-space location of self-intersections for Brownian paths. If n = d = 1, then M_λ(dt, dz) = λ(dz) N_z(dt) where N₂ is the local time at z. In the case n = 2, the set Γ can be identified with the set of Brownian loops. The measure M_λ “explodes” on the diagonal {t₁ = t₂} and, to study small loops, a random distribution which regularizes M_λ is constructed.     

Hausdorff measures of the image, graph and level set of bifractional Brownian motion

Let BH,K = {BH,K(t), t ∈ R+} be a bifractional Brownian motion in Rd. This process is a selfsimilar Gaussian process depending on two parameters H and K and it constitutes a natural generalization of fractional Brownian motion (which is obtained for K = 1). The exact Hausdorff measures of the image, graph and the level set of BH,K are investigated. The results extend the corresponding results proved by Talagrand and Xiao for fractional Brownian motion.     

A note on multitype branching process with bounded immigration in random environment

In this paper, we study the total number of progeny, W, before regenerating of multitype branching process with immigration in random environment. We show that the tail probability of |W| is of order t-κ as t→∞, with κ some constant. As an application, we prove a stable law for (L-1) random walk in random environment, generalizing the stable law for the nearest random walk in random environment (see "Kesten, Kozlov, Spitzer: A limit law for random walk in a random environment. Compositio Math., 30, 145-168 (1975)").     

Branching Brownian Motion with Catalytic Branching at the Origin

We consider a branching Brownian motion in which binary fission takes place only when particles are at the origin at a rate β>0 on the local time scale. We obtain results regarding the asymptotic behaviour of the number of particles above λt at time t, for λ>0. As a corollary, we establish the almost sure asymptotic speed of the rightmost particle. We also prove a Strong Law of Large Numbers for this catalytic branching Brownian motion.     

A branching particle system approximation for nonlinear stochastic filtering

The optimal filter π = {π t,t ∈ [0,T ]} of a stochastic signal is approximated by a sequence {π n t } of measure-valued processes defined by branching particle systems in a random environment(given by the observation process).The location and weight of each particle are governed by stochastic differential equations driven by the observation process,which is common for all particles,as well as by an individual Brownian motion,which applies to this specific particle only.The branching mechanism of each particle depends on the observation process and the path of this particle itself during its short lifetime δ = n 2α,where n is the number of initial particles and α is a fixed parameter to be optimized.As n →∞,we prove the convergence of π n t to π t uniformly for t ∈ [0,T ].Compared with the available results in the literature,the main contribution of this article is that the approximation is free of any stochastic integral which makes the numerical implementation readily available.     

Annealed survival asymptotics for Brownian motion in a scaled Poissonian potential

We consider d-dimensional Brownian motion evolving in a scaled Poissonian potential βϕ⁻²(t)V, where β>0 is a constant, ϕ is the scaling function which typically tends to infinity, and V is obtained by translating a fixed non-negative compactly supported shape function to all the particles of a d-dimensional Poissonian point process. We are interested in the large t behavior of the annealed partition sum of Brownian motion up to time t under the influence of the natural Feynman–Kac weight associated to βϕ⁻²(t)V. We prove that for d⩾2 there is a critical scale ϕ and a critical constant β_c(d)>0 such that the annealed partition sum undergoes a phase transition if β crosses β_c(d). In d=1 this picture does not hold true, which can formally be interpreted that on the critical scale ϕ we have β_c(1)=0.     

On the Mean Number of Particles of a Branching Random Walk on ℤ<Superscript><Emphasis Type="Italic">d</Emphasis></Superscript> with Periodic Sources of Branching

We consider a continuous-time branching random walk on ? ^d , where the particles are born and die on a periodic set of points (sources of branching). The spectral properties of the evolution operator for the mean number of particles at an arbitrary point of ? ^d are studied. This operator is proved to have a positive spectrum, which leads to an exponential asymptotic behavior of the mean number of particles as t → ∞.     

Uniform dimension results for Gaussian random fields

Let X = {X(t), t ∈ ℝ ^N } be a Gaussian random field with values in ℝ ^d defined by

Limit theorems for occupation time fluctuations of branching systems I: Long-range dependence

We give a functional limit theorem for the fluctuations of the rescaled occupation time process of a critical branching particle system in R^d${\mathbb{R}}^d$ with symmetric α$\alpha$-stable motion and α<d<2α$\alpha <d<2\alpha$, which leads to a long-range dependence process involving sub-fractional Brownian motion. We also give an analogous result for the system without branching and d<α$d<\alpha$, which involves fractional Brownian motion. We use a space–time random field approach.     

On scaling limits and Brownian interlacements

We consider continuous time interlacements on ? ^d , d ≥ 3, and investigate the scaling limit of their occupation times. In a suitable regime, referred to as the constant intensity regime, this brings Brownian interlacements on ? ^d into play, whereas in the high intensity regime the Gaussian free field shows up instead. We also investigate the scaling limit of the isomorphism theorem of [40]. As a by-product, when d = 3, we obtain an isomorphism theorem for Brownian interlacements.     

A limit theorem for the continuous state branching process

In this paper we discuss the limit of the martingale e^-αtK_t as t→∞, where X_t is a continuous state branching process and E[X_t] = e^αt. The important case is α > 0. Necessary and sufficient conditions are given for the limit to be positive.     

The rate of quasi sure convergence in the functional limit theorem for increments of a Brownian motion

We investigate the quasi sure convergence of the functional limit for increments of a Brownian motion. The rate of quasi sure convergence in the functional limit for increments of a d-dimensional Brownian motion is derived. The main tool in the proof is large deviation and small deviation for Brownian motion in terms of (r,p)-capacity.     

Local times on curves and uniform invariance principles

Summary Sufficient conditions are given for a family of local times |L _t ^µ | ofd-dimensional Brownian motion to be jointly continuous as a function oft and . Then invariance principles are given for the weak convergence of local times of lattice valued random walks to the local times of Brownian motion, uniformly over a large family of measures. Applications included some new results for intersection local times for Brownian motions on ² and ².Research partially supported by NSF grant DMS-8822053