Asymptotic laws for regenerative compositions: gamma subordinators and the like

For a random closed set obtained by exponential transformation of the closed range of a subordinator, a regenerative composition of generic positive integer n is defined by recording the sizes of clusters of n uniform random points as they are separated by the points of . We focus on the number of parts K_n of the composition when is derived from a gamma subordinator. We prove logarithmic asymptotics of the moments and central limit theorems for K_n and other functionals of the composition such as the number of singletons, doubletons, etc. This study complements our previous work on asymptotics of these functionals when the tail of the Lévy measure is regularly varying at 0+. Research supported in part by N.S.F. Grant DMS-0071448     

Translation invariant Gibbs states for the Ising model

We prove that all the translation invariant Gibbs states of the Ising model are a linear combination of the pure phases for any . This implies that the average magnetization is continuous for . Furthermore, combined with previous results on the slab percolation threshold [B2] this shows the validity of Pisztora's coarse graining [Pi] up to the critical temperature. We would like to thank C. Pfister and Y. Velenik for very useful comments.     

An estimate on the supremum of a nice class of stochastic integrals and U-statistics

Let a sequence of iid. random variables ξ ₁, . . . ,ξ _n be given on a space with distribution μ together with a nice class of functions f(x ₁, . . . ,x _k ) of k variables on the product space For all f ∈ we consider the random integral J _n,k (f) of the function f with respect to the k-fold product of the normalized signed measure where μ _n denotes the empirical measure defined by the random variables ξ ₁, . . . ,ξ _n and investigate the probabilities for all x>0. We show that for nice classes of functions, for instance if is a Vapnik–Červonenkis class, an almost as good bound can be given for these probabilities as in the case when only the random integral of one function is considered. A similar result holds for degenerate U-statistics, too. Supported by the OTKA foundation Nr. 037886     

Inversion de l'espace et du temps des processus de Lévy stables

We proove a new duality relation between stable Lévy processes with index and those with index . This duality appears to be the trajectorial version of the duality of Zolotarev which concerns one dimensional stable laws. We give an application of this result to the behaviour of the paths at small and large times of the process ``conditioned to stay positive'. Mots-clefs: Lévy process Stable process Fluctuation identities Ladder process     

Selector processes on classes of sets

Consider the random subset X of ℕ obtained by selecting independently each integer with a probability δ. Consider a finite class of finite sets. We describe a combinatorial quantity that is of the same order as We then give a related result allowing to compute the supremum of the empirical process on a class of sets. Work partially supported by an NSF grant.     

On the disconnection of a discrete cylinder by a random walk

We investigate the large N behavior of the time the simple random walk on the discrete cylinder needs to disconnect the discrete cylinder. We show that when d≥2, this time is roughly of order N ^{2 d} and comparable to the cover time of the slice , but substantially larger than the cover timer of the base by the projection of the walk. Further we show that by the time disconnection occurs, a massive ``clogging' typically takes place in the truncated cylinders of height . These mechanisms are in contrast with what happens when d=1.     

Exceptional times and invariance for dynamical random walks

Consider a sequence of i.i.d. random variables. Associate to each X _i (0) an independent mean-one Poisson clock. Every time a clock rings replace that X-variable by an independent copy and restart the clock. In this way, we obtain i.i.d. stationary processes {X _i (t)} _{t ≥0} (i=1,2,···) whose invariant distribution is the law ν of X ₁(0). Benjamini et al. (2003) introduced the dynamical walk S _n (t)=X ₁(t)+···+X _n (t), and proved among other things that the LIL holds for n↦S _n (t) for all t. In other words, the LIL is dynamically stable. Subsequently (2004b), we showed that in the case that the X _i (0)'s are standard normal, the classical integral test is not dynamically stable. Presently, we study the set of times t when n↦S _n (t) exceeds a given envelope infinitely often. Our analysis is made possible thanks to a connection to the Kolmogorov ɛ-entropy. When used in conjunction with the invariance principle of this paper, this connection has other interesting by-products some of which we relate. We prove also that the infinite-dimensional process converges weakly in to the Ornstein–Uhlenbeck process in For this we assume only that the increments have mean zero and variance one. In addition, we extend a result of Benjamini et al. (2003) by proving that if the X _i (0)'s are lattice, mean-zero variance-one, and possess (2+ɛ) finite absolute moments for some ɛ>0, then the recurrence of the origin is dynamically stable. To prove this we derive a gambler's ruin estimate that is valid for all lattice random walks that have mean zero and finite variance. We believe the latter may be of independent interest. The research of D. Kh. is partially supported by a grant from the NSF.     

The divergence of fluctuations for shape in first passage percolation

We consider the first passage percolation model on Z ^d for d ≥ 2. In this model, we assign independently to each edge the value zero with probability p and the value one with probability 1−p. We denote by T(0, ν) the passage time from the origin to ν for ν ∈ R ^d and It is well known that if p < p _c , there exists a compact shape B _d ⊂ R ^d such that for all > 0, t B _d (1 − ) ⊂ B(t) ⊂ tB _d (1 + ) and G(t)(1 − ) ⊂ B(t) ⊂ G(t)(1 + ) eventually w.p.1. We denote the fluctuations of B(t) from tB _d and G(t) by In this paper, we show that for all d ≥ 2 with a high probability, the fluctuations F(B(t), G(t)) and F(B(t), tB _d ) diverge with a rate of at least C log t for some constant C. The proof of this argument depends on the linearity between the number of pivotal edges of all minimizing paths and the paths themselves. This linearity is also independently interesting. Research supported by NSF grant DMS-0405150     

A stochastic flow arising in the study of local times

A stochastic flow of homeomorphisms of previously studied by Bass and Burdzy [2] and Hu and Warren [4] is shown to arise in the study of the local times of Brownian motion. This leads to a new proof of the Ray-Knight theorems for the flow via the classical Ray-Knight theorems for Brownian motion.     

The theory of energy for sub-Laplacians with an application to quasi-continuity

In this paper, we provide a suitable theory for the energy where μ is a Radon measure and Γ is the fundamental solution of a sub-Laplacian on a stratified group As a significant application, we prove the quasi-continuity of superharmonic functions related to . The proofs are elementary and mostly rely on the use of appropriate mean-value formulas and mean-integral operators relevant to the Potential Theory for .     

Classification and decomposition of Quantum Markov Semigroups

We show that a QMS on a σ-finite von Neumann algebra can be decomposed as the sum of several ``sub'-semigroups corresponding to transient and recurrent projections. We discuss two applications to physical models.     

Segal-Bargmann transforms associated with finite Coxeter groups

Using a polarization of a suitable restriction map, and heat-kernel analysis, we construct a generalized Segal-Bargmann transform associated with every finite Coxeter group G on ? ^N . We find the integral representation of this transform, and we prove its unitarity. To define the Segal-Bargmann transform, we introduce a Hilbert space of holomorphic functions on with reproducing kernel equal to the Dunkl-kernel. The definition and properties of extend naturally those of the well-known classical Fock space. The generalized Segal-Bargmann transform allows to exhibit some relationships between the Dunkl theory in the Schrödinger model and in the Fock model. Further, we prove a branching decomposition of as a unitary -module and a general version of Hecke's formula for the Dunkl transform.     

Analytic discs, plurisubharmonic hulls, and non-compactness of the -Neumann operator

We show that a complex manifold M in the boundary of a smooth bounded pseudoconvex domain Ω in is an obstruction to compactness of the -Neumann operator on Ω, provided that at some point of M, the Levi form of bΩ has the maximal possible rank n−1−dim(M) (i.e. the boundary is strictly pseudoconvex in the directions transverse to M). In particular, an analytic disc is an obstruction, provided that at some point of the disc, the Levi form has only one zero eigenvalue (i.e. the eigenvalue zero has multiplicity one). We also show that a boundary point where the Levi form has only one zero eigenvalue can be picked up by the plurisubharmonic hull of a set only via an analytic disc in the boundary. Research supported in part by NSF grant number DMS-0100517.     

Orthonormal rational function vectors

Summary In this paper, we develop a matrix framework to solve the problem of finding orthonormal rational function vectors with prescribed poles with respect to a certain discrete inner product that is defined by a set of data points and corresponding weight vectors w_i,j. Our algorithm for solving the problem is recursive, and it is of complexity If all data points are real or lie on the unit circle, then the complexity is reduced by an order of magnitude.     

Spectral multipliers for sub-Laplacians with drift on Lie groups

We study spectral multipliers of right invariant sub-Laplacians with drift on a connected Lie group G. The operators we consider are self-adjoint with respect to a positive measure , whose density with respect to the left Haar measure λ_G is a nontrivial positive character of G. We show that if p≠2 and G is amenable, then every spectral multiplier of extends to a bounded holomorphic function on a parabolic region in the complex plane, which depends on p and on the drift. When G is of polynomial growth we show that this necessary condition is nearly sufficient, by proving that bounded holomorphic functions on the appropriate parabolic region which satisfy mild regularity conditions on its boundary are spectral multipliers of . Work partially supported by the EC HARP Network “Harmonic Analysis and Related Problems”, the Progetto Cofinanziato MURST “Analisi Armonica” and the Gruppo Nazionale INdAM per l'Analisi Matematica, la Probabilità e le loro Applicazioni. Part of this work was done while the second and the third author were visiting the “Centro De Giorgi” at the Scuola Normale Superiore di Pisa, during a special trimester in Harmonic Analysis. They would like to express their gratitude to the Centro for the hospitality.     

Invertible modules for commutative -algebras with residue fields

The aim of this note is to understand under which conditions invertible modules over a commutative -algebra in the sense of Elmendorf, Kriz, Mandell & May give rise to elements in the algebraic Picard group of invertible graded modules over the coefficient ring by taking homotopy groups. If a connective commutative -algebra R has coherent localizations for every maximal ideal , then for every invertible R-module U, U_*=π_*U is an invertible graded R_*-module. In some non-connective cases we can carry the result over under the additional assumption that the commutative -algebra has ‘residue fields’ for all maximal ideals if the global dimension of R_* is small or if R is 2-periodic with underlying Noetherian complete local regular ring R₀. We apply these results to finite abelian Galois extensions of Lubin-Tate spectra.     

Reflected Symmetric α-Stable Processes and Regional Fractional Laplacian

In this paper we investigate the reflected symmetric α-stable processes and their generators. We show that the generators are regional fractional Laplacians on the closed region. In the case of 1≤α<2 their existence requires that on the boundary. Among other things we obtain the integration by parts formula of the regional fractional Laplacian and the semi-martingale decomposition of the reflected symmetric α-stable processes.     

On a class of weak Tchebycheff systems

In this paper we study the approximation power, the existence of a normalized B-basis and the structure of a degree-raising process for spaces of the formrequiring suitable assumptions on the functions u and v. The results about degree raising are detailed for special spaces of this form which have been recently introduced in the area of CAGD.     

Turbulence Phenomena in Real Analysis

The purpose of this paper is first to show that if X is any locally compact but not compact perfect Polish space and stands for the one-point compactification of X, while E^X is the equivalence relation which is defined on the Polish group C(X,R₊*) by where f, g are in C(X,R₊*), then E^X is induced by a turbulent Polish group action. Second we show that given any if we identify the n-dimensional unit sphere Sⁿ with the one-point compactification of Rⁿ via the stereographic projection, while E_n,r is the equivalence relation which is defined on the Polish group C^r(Rⁿ,R₊*) by where f, g are in C^r(Rⁿ,R₊*), then E_n,r is also induced by a turbulent Polish group action. Dedicated to my sister Alexandra and to her daughter Marianthi.     

On the category of modules of Gorenstein dimension zero

Let R be a commutative noetherian henselian non-Gorenstein local ring of depth zero. Denote by modR the category of all finitely generated R-modules, and by the full subcategory of modR consisting of all R-modules of Gorenstein dimension zero. We prove in this paper that if contains a non-free module, then it is not precovering in modR, in particular, there exist infinitely many isomorphism classes of indecomposable R-modules of Gorenstein dimension zero.