We derive properties concerning all intersection exponents for planar Brownian motion and we define generalized exponents that, loosely speaking, correspond to noninteger numbers of Brownian paths. Some of these properties lead to general conjectures concerning the exact value of these exponents.

Different extensions of an isomorphism theorem of Dynkin are developed and are used to study two distinct but related families of functionals of Lévy processes; $n$-fold “near-intersections” of a single Lévy process and continuous additive functionals of several independent Lévy processes. Intersection local times for $n$ independent Lévy processes are also studied. They are related to both of the above families. In all three cases sufficient conditions are obtained for the almost sure continuity of these functionals in terms of the almost sure continuity of associated Gaussian chaos processes. Concrete suffcient conditions are given for the almost sure continuity of these functionals of Lévy processes.

Let $X_t$ be reflected Brownian motion (RBM) in a cone with radially homogeneous reflection, killed upon reaching the vertex of the cone. We determine the adjoint process and use it to find the Martin boundary of the killed RBM together with all the corresponding positive harmonic functions. Then we can identify and prove uniqueness (up to positive scalar multiples) of the invariant measure for killed RBM and RBM without killing. Along the way, we prove the strong Feller property of the resolvent of RBM (no killing).

We provide sharp exponential moment bounds for (Stratonovich) iterated stochastic integrals under conditioning by certain small balls, including balls in certain Hölder-like norms of exponent greater than 1 /3. The proof uses a control of the variation of the Lévy area,under conditioning. The results are applied to the computation of the Onsager–Machlup functional of diffusion processes with constant diffusion matrix.

We turn the Navier-Stokes equations for a 2-dimensional viscous incompressible fluid into a system of functional integrals in the trajectory space of a suitable diffusion process. Using probabilistic techniques as Girsanov’s transformation and Bismut-Elworthy formula, we prove the existence of a unique global solution of this system in a constructive way.

We study occurrence and properties of percolation of occupied bonds in systems withrandom interactions and, hence, frustration.We develop a general argument, somewhat like Peierls’ argument, by which we show that in $\mathbb{Z},d\geq2$, percolation occurs for all possible interactions (provided they are bounded away from zero) if the parameter $p \in 1$, regulating the density of occupied bonds, is high enough. If the interactions are i.i.d. random variables then we determine bounds on the values of $p$ for which percolation occurs for all, almost all but not all, almost none but some, or none of the interactions. Motivations of this work come from the rigorous analysis of phase transitions in frustrated statistical mechanics systems.

We show that when percolation produces infinitely many infinite clusters on a Cayley graph, one cannot distinguish the clusters from each other by any invariantly defined property. This implies that uniqueness of the infinite cluster is equivalent to nondecay of connectivity (a.k.a. long-range order). We then derive applications concerning uniqueness in Kazhdan groups and in wreath products and inequalities for $p_u$.

We consider the problem of bootstrap percolation on a three-dimensional lattice and we study its finite size scaling behavior. Bootstrap percolation is an example of cellular automata defined on the $d$-dimensional lattice ${1,2,\ldots,L^d}$ in which each site can be empty or occupied by a single particle; in the starting configuration each site is occupied with probability $p$,occupied sites remain occupied forever, while empty sites are occupied by a particle if at least $\ell$ among their 2$d$ nearest neighbor sites are occupied. When $d$ is fixed, the most interesting case is the one $\ell=d:$ this is a sort of threshold, in the sense that the critical probability $p_c$ for the dynamics on the infinite lattice $\mathbb{Z}^d$ switches from zero to one when this limit is crossed. Finite size effects in the three-dimensional case are already known in the cases $\ell\leq2$; in this paper we discuss the case $\ell=3$ and we show that the finite size scaling function for this problem is of the form $f(L) =const/lnlnL$.We prove a conjecture proposed by A.C.D. van Enter.

We derive a law of large numbers for a class of multidimensional random walks in random environment satisfying a condition which first appeared in the work of Kalikow. The approach is based on the existence of a renewal structure under an assumption of “transience in the direction $l$ .” This extends, to a multidimensional context, previous work of Kesten. Our results also enable proving the convergence of the law of the environment viewed from the particle toward a limiting distribution.

For each finite measure $\Lambda$ on [0,1] a coalescent Markov process, with state space the compact set of all partitions of the set $\mathbb{N}$ of positive integers, is constructed so the restriction of the partition to each finite subset of $\mathbb{N}$ is a Markov chain with the following transition rates: when the partition has b blocks, each $k$-tuple of blocks is merging to form a single block at rate $\int_0^1x^{k-2}(1-x)^{b-k}\Lambda(dx)$. Call this process a $\Lambda$-coalescent. Discrete measure-valued processes derived from the $\Lambda$-coalescent model a system of masses undergoing coalescent collisions. Kingman’s coalescent, which has numerous applications in population genetics, is the $\delta_0$-coalescent for $\delta_0$ a unit mass at 0. The coalescent recently derived by Bolthausen and Sznitman from Ruelle’s probability cascades, in the context of the Sherrington–Kirkpatrick spin glass model in mathematical physics, is the $U$-coalescent for $U$ uniform on [0,1]. For $\Lambda=U$, and whenever an infinite number of masses are present, each collision in $\Lambda$-coalescent involves an infinite number of masses almost surely, and the proportion of masses involved exists as a limit almost surely and is distributed proportionally to $\Lambda$. The two-parameter Poisson–Dirichlet family of random discrete distributions derived from a stable subordinator, and corresponding exchangeable random partitions of $\mathbb{N}$ governed by a generalization of the Ewens sampling formula, are applied to describe transition mechanisms for processes of coalescence and fragmentation, including the $U$-coalescent and its time reversal.

We discuss an approach, based on the Brunn–Minkowski inequality, to isoperimetric and analytic inequalities for probability measures on Euclidean space with logarithmically concave densities. In particular, we show that such measures have positive isoperimetric constants in the sense of Cheeger and thus always share Poincaré-type inequalities. We then describe those log-concave measures which satisfy isoperimetric inequalities of Gaussian type. The results are precised in dimension 1.

We prove that the inequality $\Psi^-1(\mu(tA))\geq t\Psi^-1(\mu(A))$ holds for any centered Gaussian measure $\mu$ on a separable Banach space $F$, any convex, closed, symmetric set $A\subset{F}$ and $t\geq1$, where $\Psi(x)=\gamma_1(-x,x)=(2\pi)^-1/2\int_{-x}^x\exp(-y^2)2)dy$. As an application, the best constants in comparison of moments of Gaussian vectors are calculated.

Our purpose is to prove a particular case of a conjecture concerning the Gaussian measure of the intersection of two symmetric convex sets of $\mathbb{R}^n$. This conjecture states that the measure of the intersection is greater or equal to the product of the measures. In this paper, we prove the inequality when one of the two convex sets is a symmetric ellipsoid and the other one is simply symmetric. The general case is still open.

An iterated function system on $\mathscr{X}\subset\mathbb{R}^d$ is defined by successively applying an i.i.d.sequence of random Lipschitz functions from to $\mathscr{X}$ to $\mathscr{X}$. This paper shows how $F _n = f_1\circ\ldots\circ f_n$ may converge even in the absence of the strong contraction conditions, for instance, Lipschitz constant smaller than 1 on average,which earlier work has required. Instead, it is posited that there be a region of contraction which compensates for the noncontractive or even expansive part of the functions. Applications to queues, to self-modifying random walks and to random logistic maps are given.

Let $\xi_1,\xi_2\ldots$ be a sequence of i.i.d.random variables, and consider the elementary symmetric polynomial $S ^(k)(n)$ of order $k =k(n)$ of the first $n$ elements $\xi_1\ldots,\xi_n$ of this sequence. We are interested in the limit behavior of $S^(k) (n)$ with an appropriate transformation if $k(n)/n\rightarrow\alpha, 0<\alpha<1$. Since $k(n)\rightarrow\infty$ as $n\rightarrow\infty$, the classical methods cannot be applied in this case and new kinds of results appear.We solve the problem under some conditions which are satisfied in the generic case. The proof is based on the saddlepoint method and a limit theorem for sums of independent random vectors which mayhave some special interest in itself.

The ballot theorem and the uniform law for sojourn times, both results known for cyclically stationary sequences and processes on a bounded index set, are here extended to infinite, stationary sequences and to stationary processes on $\mathbb{R}_+$. Our extensions contain all previously known versions as special cases.

For a convex body $K$ in the plane, let $p(n, K)$ denote the probability that $n$ random, independent, and uniform points from $K$ are in convex position, that is, none of them lies in the convex hull of the others. Here we determine the asymptotic behavior of $p(n, K)$ by showing that, as $n$ goes to infinity,$ {n^2}^n\sqrt{p(n,K)}$ tends to a finite and positive limit.

A stationary renewal process $N(\cdot)$ for which the lifetime distribution has its $k$th moment finite or infinite according as $k$ is less than or greater than $\kappa$ for some $1 \lt \kappa \lt 2$, is long-range dependent and has Hurst index $\alpha=1/2(3-\kappa)$ (this is the critical index $\alpha$ for which $\lim\sup_{t \to \infty} t^{-2a}$ var $N(0,t]$ is finite or infinite according as $\alpha$ is greater than or less than $\alpha$. This identification is accomplished by delineating the growth rate properties of the difference between the renewal function and its linear asymptote, thereby extending work of Täcklind.

First- and second-order laws of the iterated logarithm are obtained for both the Nelson–Aalen and the Kaplan–Meier estimators in the random censorship model, uniform up to a large order statistic of the censored data. The rates for the first-order processes are exact except for constants. The LIL for the second-order processes (where one subtracts a linear, empirical process, term from the difference between the original process and the estimator), uniform over fixed intervals, is also proved. Somewhat surprisingly, there is a certain degree of proof unification for fixed and variable intervals in the second-order results for the Nelson–Aalen estimator. No assumptions are made on the distribution of the censoring variables and only continuity of the distribution function of the original variables is assumed for the results on the Kaplan–Meier estimator.

In this paper we shall derive exponential nonuniformBerry–Esseen bounds in the central limit theorem for self-normalized sums.We show that the size of the error can be reduced considerably by replacing the usual standardization by self-normalization. In particular, we establish the exponential bounds for the probability of the self-normalized sums under the condition that the third moment is finite, whereas an exponential moment assumption is required for the standardized sums. Applications to t-statistics and the probabilities of moderate deviations of self-normalized sums are also discussed.

A general and precise Berry –Esseen bound is proved for the Studentized mean based on $N$ random observations drawn without replacement from a finite population. The bound yields the optimal rate $O(N^ -1 /2)$ under minimal conditions. If the Erdős–Rényi condition holds this bound implies the asymptotic normality of Student’s statistic and the self-normalized sum.