Let Ξ be a discrete set in ℝ^d. Call the elements of Ξ centers. The well-known Voronoi tessellation partitions ℝ^d into polyhedral regions (of varying sizes) by allocating each site of ℝ^d to the closest center. Here we study “fair” allocations of ℝ^d to Ξ in which the regions allocated to different centers have equal volumes.

We prove that if Ξ is obtained from a translation-invariant point process, then there is a unique fair allocation which is stable in the sense of the Gale–Shapley marriage problem. (I.e., sites and centers both prefer to be allocated as close as possible, and an allocation is said to be unstable if some site and center both prefer each other over their current allocations.)

We show that the region allocated to each center ξ is a union of finitely many bounded connected sets. However, in the case of a Poisson process, an infinite volume of sites are allocated to centers further away than ξ. We prove power law lower bounds on the allocation distance of a typical site. It is an open problem to prove any upper bound in d>1.

We show that, for a stationary version of Hammersley’s process, with Poisson sources on the positive x-axis and Poisson sinks on the positive y-axis, the variance of the length of a longest weakly North–East path L(t,t) from (0,0) to (t,t) is equal to $2\mathbb{E}(t-X(t))_{+}$, where X(t) is the location of a second class particle at time t. This implies that both $\mathbb{E}(t-X(t))_{+}$ and the variance of L(t,t) are of order t^2/3. Proofs are based on the relation between the flux and the path of a second class particle, continuing the approach of Cator and Groeneboom [Ann. Probab. 33 (2005) 879–903].

We consider a system of asymmetric independent random walks on ℤ^d, denoted by {η_t,t∈ℝ}, stationary under the product Poisson measure ν_ρ of marginal density ρ>0. We fix a pattern $\mathcal{A}$, an increasing local event, and denote by τ the hitting time of $\mathcal{A}$. By using a loss network representation of our system, at small density, we obtain a coupling between the laws of η_t conditioned on {τ>t} for all times t. When d≥3, this provides bounds on the rate of convergence of the law of η_t conditioned on {τ>t} toward its limiting probability measure as t tends to infinity. We also treat the case where the initial measure is close to ν_ρ without being product.

We consider attractive irreducible conservative particle systems on ℤ, without necessarily nearest-neighbor jumps or explicit invariant measures. We prove that for such systems, the hydrodynamic limit under Euler time scaling exists and is given by the entropy solution to some scalar conservation law with Lipschitz-continuous flux. Our approach is a generalization of Bahadoran et al. [Stochastic Process. Appl. 99 (2002) 1–30], from which we relax the assumption that the process has explicit invariant measures.

We introduce two probabilistic models for N interacting Brownian motions moving in a trap in ℝ^d under mutually repellent forces. The two models are defined in terms of transformed path measures on finite time intervals under a trap Hamiltonian and two respective pair-interaction Hamiltonians. The first pair interaction exhibits a particle repellency, while the second one imposes a path repellency.

We analyze both models in the limit of diverging time with fixed number N of Brownian motions. In particular, we prove large deviations principles for the normalized occupation measures. The minimizers of the rate functions are related to a certain associated operator, the Hamilton operator for a system of N interacting trapped particles. More precisely, in the particle-repellency model, the minimizer is its ground state, and in the path-repellency model, the minimizers are its ground product-states. In the case of path-repellency, we also discuss the case of a Dirac-type interaction, which is rigorously defined in terms of Brownian intersection local times. We prove a large-deviation result for a discrete variant of the model.

This study is a contribution to the search for a mathematical formulation of the quantum system of N trapped interacting bosons as a model for Bose–Einstein condensation, motivated by the success of the famous 1995 experiments. Recently, Lieb et al. described the large-N behavior of the ground state in terms of the well-known Gross–Pitaevskii formula, involving the scattering length of the pair potential. We prove that the large-N behavior of the ground product-states is also described by the Gross–Pitaevskii formula, however, with the scattering length of the pair potential replaced by its integral.

We study the hitting properties of the solutions u of a class of parabolic stochastic partial differential equations with singular drifts that prevent u from becoming negative. The drifts can be a reflecting term or a nonlinearity cu⁻³, with c>0. We prove that almost surely, for all time t>0, the solution u_t hits the level 0 only at a finite number of space points, which depends explicitly on c. In particular, this number of hits never exceeds 4 and if c>15/8, then level 0 is not hit.

A formula for the transition density of a Markov process defined by an infinite-dimensional stochastic equation is given in terms of the Ornstein–Uhlenbeck bridge and a useful lower estimate on the density is provided. As a consequence, uniform exponential ergodicity and V-ergodicity are proved for a large class of equations. We also provide computable bounds on the convergence rates and the spectral gap for the Markov semigroups defined by the equations. The bounds turn out to be uniform with respect to a large family of nonlinear drift coefficients. Examples of finite-dimensional stochastic equations and semilinear parabolic equations are given.

We solve an open problem of Diaconis that asks what are the largest orders of p_n and q_n such that Z_n, the p_n×q_n upper left block of a random matrix Γ_n which is uniformly distributed on the orthogonal group O(n), can be approximated by independent standard normals? This problem is solved by two different approximation methods.

First, we show that the variation distance between the joint distribution of entries of Z_n and that of p_nq_n independent standard normals goes to zero provided $p_{n}=o(\sqrt{n}\,)$ and $q_{n}=o(\sqrt{n}\,)$. We also show that the above variation distance does not go to zero if $p_{n}=[x\sqrt{n}\,]$ and $q_{n}=[y\sqrt{n}\,]$ for any positive numbers x and y. This says that the largest orders of p_n and q_n are o(n^1/2) in the sense of the above approximation.

Second, suppose Γ_n=(γ_ij)_n×n is generated by performing the Gram–Schmidt algorithm on the columns of Y_n=(y_ij)_n×n, where {y_ij;1≤i,j≤n} are i.i.d. standard normals. We show that $\varepsilon _{n}(m):=\max_{1\leq i\leq n,1\leq j\leq m}|\sqrt{n}\cdot\gamma_{ij}-y_{ij}|$ goes to zero in probability as long as m=m_n=o(n/logn). We also prove that $\varepsilon _{n}(m_{n})\to 2\sqrt{\alpha}$ in probability when m_n=[nα/logn] for any α>0. This says that m_n=o(n/logn) is the largest order such that the entries of the first m_n columns of Γ_n can be approximated simultaneously by independent standard normals.

Dunkl processes are martingales as well as càdlàg homogeneous Markov processes taking values in ℝ^d and they are naturally associated with a root system. In this paper we study the jumps of these processes, we describe precisely their martingale decompositions into continuous and purely discontinuous parts and we obtain a Wiener chaos decomposition of the corresponding L² spaces of these processes in terms of adequate mixed multiple stochastic integrals.

Soit (ɛ_n)_n∈Z un jeu de pile ou face, c’est-à-dire une suite de variables aléatoires indépendantes de loi (δ₋₁+δ₁)/2, et (H_n)_n∈Z un processus à valeurs dans {−1,1}, prévisible dans la filtration naturelle de (ɛ_n)_n∈Z. Alors (H_nɛ_n)_n∈Z est encore un jeu de pile ou face, dont la filtration naturelle est contenue dans celle de (ɛ_n)_n∈Z. Le but de l’article est d’obtenir des conditions pour que ces filtrations soient égales et de décrire l’écart entre ces filtrations lorsqu’elles sont différentes. Nous nous intéressons plus particulièrement au cas des transformations homogènes, où le processus (H_nɛ_n)_n∈Z est une fonctionnelle de (ɛ_n)_n∈Z qui commute avec les translations. Nous étudions de façon approfondie les transformations homogènes de longueur finie, où H_n est de la forme ϕ(ɛ_n−d,…,ɛ_n−1) avec d∈N et ϕ:{−1;1}^d→{−1;1} fixés.

We study the standard property of the natural filtration associated to a 0–1 valued stationary process. In our main result we show that if the process has summable memory decay, then the associated filtration is standard. We prove it by coupling techniques. For a process whose associated filtration is standard, we construct a product type filtration extending it, based upon the usual couplings and the Vershik’s criterion for standardness.

Cramér and Leadbetter introduced in 1967 the sufficient condition $$\frac{r''(s)-r''(0)}{s}\in L^{1}([0,\delta],dx),\qquad \delta>0,$$ to have a finite variance of the number of zeros of a centered stationary Gaussian process with twice differentiable covariance function r. This condition is known as the Geman condition, since Geman proved in 1972 that it was also a necessary condition. Up to now no such criterion was known for counts of crossings of a level other than the mean. This paper shows that the Geman condition is still sufficient and necessary to have a finite variance of the number of any fixed level crossings. For the generalization to the number of a curve crossings, a condition on the curve has to be added to the Geman condition.

We establish the central limit theorem for linear processes with dependent innovations including martingales and mixingale type of assumptions as defined in McLeish [Ann. Probab. 5 (1977) 616–621] and motivated by Gordin [Soviet Math. Dokl. 10 (1969) 1174–1176]. In doing so we shall preserve the generality of the coefficients, including the long range dependence case, and we shall express the variance of partial sums in a form easy to apply. Ergodicity is not required.

Let u be a pluriharmonic function on the unit ball in ℂⁿ. I consider the relationship between the set of points L_u on the boundary of the ball at which u converges nontangentially and the set of points ℒ_u at which u converges along conditioned Brownian paths. For harmonic functions u of two variables, the result $L_{u}\stackrel{\mathrm{a.e.}}{=}\mathscr{L}_{u}$ has been known for some time, as has a counterexample to the same equality for three variable harmonic functions. I extend the $L_{u}\stackrel{\mathrm{a.e.}}{=}\mathscr{L}_{u}$ result to pluriharmonic functions in arbitrary dimensions.

Khoshnevisan and Xiao showed in [Ann. Probab. 33 (2005) 841–878] that the statement about almost surely vanishing Bessel–Riesz capacity of the image of a Borel set G⊂ℝ₊ under a symmetric Lévy process X in ℝ^d is equivalent to the vanishing of a deterministic f-capacity for a particular function f defined in terms of the characteristic exponent of X. The authors conjectured that a similar statement is true for all Lévy processes in ℝ^d. We show that the conjecture is true provided we extend the definition of f and require certain integrability conditions which cannot be avoided in general.

Estimate (3.39) which appears in the proof of Proposition 3.4 in [Ann. Probab. 27 (1999) 1414–1467] is wrong. We present below a corrected proof which introduces an extra factor 2 in equations (3.34) and (3.35). This has no consequence in the rest of the paper since Proposition 3.4 is used to estimate only ratios; see (3.23) and (3.25).