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.     

Path continuity of fractional Dirichlet functionals

We prove that the quasi continuous version of a functional in E^p_r is continuous along the sample paths of the Dirichlet process provided that p>2, 0<r?1 and pr>2, without assuming the Meyer equivalence. Parallel results for multi-parameter processes are also obtained. Moreover, for 1<p<2, we prove that a n parameter Dirichlet process does not touch a set of (p,2n)-zero capacity. As an example, we also study the quasi-everywhere existence of the local times of martingales on path space.     

Comparisons of relative BV-capacities and Sobolev capacity in metric spaces

We study relations between the variational Sobolev 1-capacity and versions of variational BV-capacity in a complete metric space equipped with a doubling measure and supporting a weak (1,1)-Poincaré inequality. We prove the equality of 1-modulus and the continuous 1-capacity, extending the known results for 1<p<∞ to also cover the more geometric case p=1. Then we give alternative definitions for variational BV-capacities and obtain equivalence results between them. Finally we study relations between total 1-capacity and versions of BV-capacity.     

Capacities, surface area, and radial sums

A dual capacitary Brunn-Minkowski inequality is established for the (n−1)-capacity of radial sums of star bodies in Rn. This inequality is a counterpart to the capacitary Brunn-Minkowski inequality for the p-capacity of Minkowski sums of convex bodies in Rn, 1?p<n, proved by Borell, Colesanti, and Salani. When n?3, the dual capacitary Brunn-Minkowski inequality follows from an inequality of Bandle and Marcus, but here a new proof is given that provides an equality condition. Note that when n=3, the (n−1)-capacity is the classical electrostatic capacity. A proof is also given of both the inequality and a (different) equality condition when n=2. The latter case requires completely different techniques and an understanding of the behavior of surface area (perimeter) under the operation of radial sum. These results can be viewed as showing that in a sense (n−1)-capacity has the same status as volume in that it plays the role of its own dual set function in the Brunn-Minkowski and dual Brunn-Minkowski theories.     

Limit distributions for the maxima of stationary Gaussian processes

Let {X_n} be a stationary Gaussian sequence with E{X₀} = 0, {X²₀} = 1 and E{X₀X_n} = rⁿ_n Let c_n = (2ln n)^{$\text{built1}\text{2}$}, b_n = c_n? $\text{1}\text{2}$c^-1_n ln(4π ln n), and set M_n = max_{0 ?k?n}X_k. A classical result for independent normal random variables is that

$\text{P[c}{}_{\mathrm{n}}\text{(M}{}_{\mathrm{n}}\text{?b}{}_{\mathrm{n}}\text{)?x]→exp[-e}{}^{\mathrm{-x}}\text{] as n → ∞ for all x.}$

Berman has shown that (1) applies as well to dependent sequences provided r_nlnn = o(1). Suppose now that {r_n} is a convex correlation sequence satisfying r_n = o(1), (r_nlnn)^-1 is monotone for large n and o(1). Then

$\text{P[r}{}_{\mathrm{n}}{}^{\mathrm{<mtext>-1</mtext><mtext>2</mtext>}}\text{(M}{}_{\mathrm{n}}\text{? (1?r}{}_{\mathrm{n}}\text{)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{b}{}_{\mathrm{n}}\text{)?x] → Ф(x)}$

for all x, where Ф is the normal distribution function. While the normal can thus be viewed as a second natural limit distribution for {M_n}, there are others. In particular, the limit distribution is given below when r_n is (sufficiently close to) γ/ln n. We further exhibit a collection of limit distributions which can arise when r_n decays to zero in a nonsmooth manner. Continuous parameter Gaussian processes are also considered. A modified version of (1) has been given by Pickands for some continuous processes which possess sufficient asymptotic independence properties. Under a weaker form of asymptotic independence, we obtain a version of (2).     

Topological stability through extremely tame retractions

Suppose that F:(Rn×Rd,0)→(Rp×Rd,0) is a smoothly stable, Rd-level preserving germ which unfolds f:(Rn,0)→(Rp,0); then f is smoothly stable if and only if we can find a pair of smooth retractions r:(Rn+d,0)→(Rn,0) and s:(Rp+d,0)→(Rp,0) such that f°r=s°F. Unfortunately, we do not know whether f will be topologically stable if we can find a pair of continuous retractions r and s.The class of extremely tame (E-tame) retractions, introduced by du Plessis and Wall, are defined by their nice geometric properties, which are sufficient to ensure that f is topologically stable.In this article, we present the E-tame retractions and their relation with topological stability, survey recent results by the author concerning their construction, and illustrate the use of our techniques by constructing E-tame retractions for certain germs belonging to the E- and Z-series of singularities.     

The influence of a log-type small ball factor in the study of the lower classes

Let {BH₁,H₂(t₁,t₂),t₁?0,t₂?0} be a fractional Brownian sheet with indexes 0<H₁,H₂<1. When H₁=H₂:=H, there is a logarithmic factor in the small ball function of the sup-norm statistic of BH,H. First, we state general conditions (one based on a logarithmic factor in the small ball function) on some statistics of BH,H. Then we characterize the sufficiency part of the lower classes of these statistics by an integral test. Finally, when we consider the sup-norm statistic, the influence of the log-type small ball factor in the necessity part is measured by a second integral test.     

Estimates and asymptotic expansions for condenser p-capacities. The anisotropic case of segments

We provide estimates and asymptotic expansions of condenser p-capacities and focus on the anisotropic case of (line) segments.

After preliminary results, we study p-capacities of points with respect to asymptotic approximations, positivity cases and convergence speed of descending continuity. We introduce equidistant condensers to point out that the anisotropy caused by a segment in the p-Laplace equation is such that the Pólya-Szegö rearrangement inequality for Dirichlet type integrals yields a trivial lower bound. Moreover, when p > N , one cannot build an admissible solution for a segment, however small its length may be, by extending the case of a punctual obstacle.

Our main contribution is to provide a lower bound to the N -dimensional condenser p-capacity of a segment, by means of the N -dimensional and of the (N ?1)-dimensional condenser p-capacities of a point. The positivity cases follow for p-capacities of segments. Our method could be extended to obstacles with codimensions ≥ 2 in higher dimensions, such as surfaces in ?⁴.

Introducing elliptical condensers, we obtain an estimate and the asymptotic expansion for the condenser 2-capacity of a segment in the plane. The topological gradient of the 2-capacity is not an appropriate tool to separate curves and obstacles with non-empty interior in 2D. In the case p ≠ 2, elliptical condensers should prove useful to obtain further estimates of p-capacities of segments.     

The Frobenius map, rank 2 vector bundles and Kummer's quartic surface in characteristic 2 and 3

Let X be a smooth projective curve of genus g?2 defined over an algebraically closed field k of characteristic p>0. Let M_X(r) be the moduli space of semi-stable vector bundles with fixed trivial determinant. The relative Frobenius map induces by pull-back a rational map . We determine the equations of V in the following two cases (1) (g,r,p)=(2,2,2) and X nonordinary with Hasse-Witt invariant equal to 1 (see math.AG/0005044 for the case X ordinary), and (2) (g,r,p)=(2,2,3). We also show the existence of base points of V, i.e., semi-stable bundles E such that F∗E is not semi-stable, for any triple (g,r,p).     

On the Gaussian approximation of vector-valued multiple integrals

By combining the findings of two recent, seminal papers by Nualart, Peccati and Tudor, we get that the convergence in law of any sequence of vector-valued multiple integrals F_n towards a centered Gaussian random vector N, with given covariance matrix C, is reduced to just the convergence of: (i) the fourth cumulant of each component of F_n to zero; (ii) the covariance matrix of F_n to C. The aim of this paper is to understand more deeply this somewhat surprising phenomenon. To reach this goal, we offer two results of a different nature. The first one is an explicit bound for d(F,N) in terms of the fourth cumulants of the components of F, when F is a R^d-valued random vector whose components are multiple integrals of possibly different orders, N is the Gaussian counterpart of F (that is, a Gaussian centered vector sharing the same covariance with F) and d stands for the Wasserstein distance. The second one is a new expression for the cumulants of F as above, from which it is easy to derive yet another proof of the previously quoted result by Nualart, Peccati and Tudor.     

Continuity of l 2 two-parameter ornstein-uhlenbeck processes

Let {Y(t);t=(t ₁,t₂)≥0}={X_k(t₁,t₂);t₁≥0,t₂≥0} _k=1 ^∞ , be a sequence of two-parameter Ornstein-Uhlenbeck processes (OUP₂) with coefficient a _k>0,ß_k>0.. A Fernique type inequality is established and the sufficient condition for a. s. l ² continuity of Y(?) is studied by means of the inequality.     

Processus d'Ornstein-Uhlenbeck et ensemblesW 2,2—polaires

Résumé Nous démontrons que, dans un espace de Fréchet gaussien (E,), les ensembles dec _2,2-capacité nulle sont les ensembles que le processus d'Ornstein-Uhlenbeck à deux indices à valeurs dansE ne rencontres pas. Afin de prouver ce résultat, nous établissons une inégalité entre la probabilité de rencontre d'un ensemble et la capacité de celui-ci. Nous établissons également une représentation de lac _2,2-capacité d'un ensemble par une mesure aléatoire à deux indices.

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We show that, on a Fréchet gaussian space (E,) thec _2,2-capacity null sets are those which cannot be hit by the two-parameterE-valued Ornstein-Uhlenbeck process. In order to prove this result, an inequality between the hitting probability of a set and itsc _2,2-capacity, as well as a representation of thec _2,2-capacity of a set by a two parameter random measure, is established.

     

Extremes of conditioned elliptical random vectors

Let {X_n,n?1} be iid elliptical random vectors in R^d,d≥2 and let I,J be two non-empty disjoint index sets. Denote by X_n,I,X_n,J the subvectors of X_n with indices in I,J, respectively. For any a∈R^d such that a_J is in the support of X_1,J the conditional random sample X_n,I|X_n,J=a_J,n≥1 consists of elliptically distributed random vectors. In this paper we investigate the relation between the asymptotic behaviour of the multivariate extremes of the conditional sample and the unconditional one. We show that the asymptotic behaviour of the multivariate extremes of both samples is the same, provided that the associated random radius of X₁ has distribution function in the max-domain of attraction of a univariate extreme value distribution.     

Weakly monotone functions

The definition of monotone function in the sense of Lebesgue is extended to the Sobolev spacesW ^1,p ,p >n ? 1. It is proven that such weakly monotone functions are continuous except in a singular set ofp-capacity zero that is empty in the casep =n. Applications to the regularity of mappings with finite dilatation appearing in nonlinear elasticity theory are given.     

On diffusivity of a tagged particle in asymmetric zero-range dynamics

Consider a distinguished, or tagged particle in zero-range dynamics on Zd with rate g whose finite-range jump probabilities p possess a drift ∑jp(j)≠0. We show, in equilibrium, that the variance of the tagged particle position at time t is at least order t in all d?1, and at most order t in d=1 and d?3 for a wide class of rates g. Also, in d=1, when the jump distribution p is totally asymmetric and nearest-neighbor, and the rate g(k) increases, and g(k)/k either decreases or increases with k, we show the diffusively scaled centered tagged particle position converges to a Brownian motion with a homogenized diffusion coefficient in the sense of finite-dimensional distributions. Some characterizations of the tagged particle variance are also given.     

p-Adic liftings of the supersingular j-invariants and j-zeros of certain Eisenstein series

Let p>3 be a prime. We consider j-zeros of Eisenstein series Ek of weights k=p−1+Mpa(p²−1) with M,a?0 as elements of . If M=0, the j-zeros of Ep−1 belong to Qp(ζp²−1) by Hensel's lemma. Call these j-zeros p-adic liftings of supersingular j-invariants. We show that for every such lifting u there is a j-zero r of Ek such that ordp(r−u)>a. Applications of this result are considered. The proof is based on the techniques of formal groups.     

Moments, moderate and large deviations for a branching process in a random environment

Let (Z_n) be a supercritical branching process in a random environment ξ, and W be the limit of the normalized population size Z_n/E[Z_n|ξ]. We show large and moderate deviation principles for the sequence logZ_n (with appropriate normalization). For the proof, we calculate the critical value for the existence of harmonic moments of W, and show an equivalence for all the moments of Z_n. Central limit theorems on W−W_n and logZ_n are also established.     

Empirical processes of multidimensional systems with multiple mixing properties

We establish a multivariate empirical process central limit theorem for stationary R^d-valued stochastic processes (X_i)_i≥1 under very weak conditions concerning the dependence structure of the process. As an application, we can prove the empirical process CLT for ergodic torus automorphisms. Our results also apply to Markov chains and dynamical systems having a spectral gap on some Banach space of functions. Our proof uses a multivariate extension of the techniques introduced by Dehling et al. (2009) [9] in the univariate case. As an important technical ingredient, we prove a 2pth moment bound for partial sums in multiple mixing systems.