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.     

Convergence to fractional kinetics for random walks associated with unbounded conductances

We consider a random walk among unbounded random conductances whose distribution has infinite expectation and polynomial tail. We prove that the scaling limit of this process is a Fractional-Kinetics process??that is the time change of a d-dimensional Brownian motion by the inverse of an independent ??-stable subordinator. We further show that the same process appears in the scaling limit of the non-symmetric Bouchaud??s trap model.     

A central limit theorem for branching Brownian motion with random immigration

We establish a central limit theorem for a branching Brownian motion with random immigration under the annealed law,where the immigration is determined by another branching Brownian motion.The limit is a Gaussian random measure and the normalization is t3/4for d=3 and t1/2for d≥4,where in the critical dimension d=4 both the immigration and the branching Brownian motion itself make contributions to the covariance of the limit.     

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.     

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.     

The discrete Gaussian free field on a compact manifold

In this article we define the discrete Gaussian free field (DGFF) on a compact manifold. Since there is no canonical grid approximation of a manifold, we construct a random graph that suitably replaces the square lattice ${\mathbb{Z}}^d$ in Euclidean space, and prove that the scaling limit of the DGFF is given by the manifold continuum Gaussian free field (GFF). Furthermore using Voronoi tessellations we can interpret the DGFF as element of a Sobolev space and show convergence to the GFF in law with respect to the strong Sobolev topology.     

On the Central Limit Theorem for Nonuniform φ-Mixing Random Fields

The partial-sum processes, indexed by sets, of a stationary nonuniform -mixing random field on the d-dimensional integer lattice are considered. A moment inequality is given from which the convergence of the finite-dimensional distributions to a Brownian motion on the Borel subsets of [0, 1] ^d is obtained. A Uniform CLT is proved for classes of sets with a metric entropy restriction and applied to certain Gibbs fields. This extends some results of Chen⁽⁵⁾ for rectangles. In this case and when the variables are bounded a simpler proof of the uniform CLT is given.     

A heavy traffic limit theorem for a class of open queueing networks with finite buffers

We consider a queueing network of d single server stations. Each station has a finite capacity waiting buffer, and all customers served at a station are homogeneous in terms of service requirements and routing. The routing is assumed to be deterministic and hence feedforward. A server stops working when the downstream buffer is full. We show that a properly normalized d-dimensional queue length process converges in distribution to a fd-dimensional semimartingale reflecting Brownian motion (RBM) in a d-dimensional box under a heavy traffic condition. The conventional continuous mapping approach does not apply here because the solution to our Skorohod problem may not be unique. Our proof relies heavily on a uniform oscillation result for solutions to a family of Skorohod problems. The oscillation result is proved in a general form that may be of independent interest. It has the potential to be used as an important ingredient in establishing heavy traffic limit theorems for general finite buffer networks. This revised version was published online in June 2006 with corrections to the Cover Date.     

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

Kahane using brownian local times of intersection

We show that if a set E in the positive real line has Hausdorff dimension greater than d/2 m, then the m-fold algebraic sum of the image of E by d-dimensional Brownian motion has an interior point. This extends a result of Kahane. The proof uses techniques found in Rosen (1983) and Geman, Horowitz and Rosen. We then show that the results do not hold for random sets and demonstrate that the above condition on the Hausdorff dimension of E is not close to being necessary     

Sharp Asymptotics for the Partition Function of Some Continuous-time Directed Polymers

This paper is concerned with two related types of directed polymers in a random medium. The first one is a d-dimensional Brownian motion living in a random environment which is white-noise in time and homogeneous in space. The second is a continuous-time discrete-space Markov process on ℤ ^d , in a random environment with similar properties as in continuous space, albeit defined only on . The case of a space-time white noise environment can be achieved in this second setting. By means of some Gaussian tools, we estimate the free energy of these models at low temperature, and give some further information on the strong disorder regime of the objects under consideration. Frederi Viens’ research partially supported by NSF grants no.: DMS 0204999 and DMS 0606615.     

Phase transition of the principal Dirichlet eigenvalue in a scaled Poissonian potential

We consider d-dimensional Brownian motion in a scaled Poissonian potential and the principal Dirichlet eigenvalue (ground state energy) of the corresponding Schr?dinger operator. The scaling is chosen to be of critical order, i.e. it is determined by the typical size of large holes in the Poissonian cloud. We prove existence of a phase transition in dimensions d≥ 4: There exists a critical scaling constant for the potential. Below this constant the scaled infinite volume limit of the corresponding principal Dirichlet eigenvalue is linear in the scale. On the other hand, for large values of the scaling constant this limit is strictly smaller than the linear bound. For d > 4 we prove that this phase transition does not take place on that scale. Further we show that the analogous picture holds true for the partition sum of the underlying motion process. Received: 10 December 1999 / Revised version: 14 July 2000/?Published online: 15 February 2001     

Integral representation of renormalized self-intersection local times

In this paper we apply Clark-Ocone formula to deduce an explicit integral representation for the renormalized self-intersection local time of the d-dimensional fractional Brownian motion with Hurst parameter H∈(0,1). As a consequence, we derive the existence of some exponential moments for this random variable.     

Regularity of the Local Time for the d-dimensional Fractional Brownian Motion with N-parameters

ABSTRACT

We give the Wiener–Ito? chaotic decomposition for the local time of the d-dimensional fractional Brownian motion with N-parameters and study its smoothness in the Sobolev–Watanabe spaces.     

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     

Tightness of the recentered maximum of the two‐dimensional discrete Gaussian free field

We consider the maximum of the discrete two‐dimensional Gaussian free field (GFF) in a box and prove that its maximum, centered at its mean, is tight, settling a longstanding conjecture. The proof combines a recent observation by Bolthausen, Deuschel, and Zeitouni with elements from Bramson's results on branching Brownian motion and comparison theorems for Gaussian fields. An essential part of the argument is the precise evaluation, up to an error of order 1, of the expected value of the maximum of the GFF in a box. Related Gaussian fields, such as the GFF on a two‐dimensional torus, are also discussed. © 2011 Wiley Periodicals, Inc.     

Functional central limit theorem for super α-stable processes

A functional central limit theorem is proved for the centered occupation time process of the super α-stable processes in the finite dimensional distribution sense. For the intermediate dimensions α < d < 2α (0 < α ≤ 2), the limiting process is a Gaussian process, whose covariance is specified; for the critical dimension d= 2α and higher dimensions d < 2α, the limiting process is Brownian motion. Zhang Mei, Functional central limit theorem for the super-brownian motion with super-Brownian immigration, J. Theoret. Probab., to appear.     

Scaling identity for crossing Brownian motion in a Poissonian potential

We consider d-dimensional Brownian motion in a truncated Poissonian potential (d≥ 2). If Brownian motion starts at the origin and ends in the closed ball with center y and radius 1, then the transverse fluctuation of the path is expected to be of order |y|^ξ, whereas the distance fluctuation is of order |y|^χ. Physics literature tells us that ξ and χ should satisfy a scaling identity 2ξ− 1 = χ. We give here rigorous results for this conjecture. Received: 31 December 1997 / Revised version: 14 April 1998     

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     

Regularity of local times of random fields

In this paper, we study the fractional smoothness of local times of general processes starting from the occupation time formula, and obtain the quasi-sure existence of local times in the sense of the Malliavin calculus. This general result is then applied to the local times of N-parameter d-dimensional Brownian motions, fractional Brownian motions and the self-intersection local time of the 2-dimensional Brownian motion, as well as smooth semimartingales.