Consider the following mechanism for the random evolution of a distribution of mass on the integer lattice Z. At unit rate, independently for each site, the mass at the site is split into two parts by choosing a random proportion distributed according to some specified probability measure on [0, 1] and dividing the mass in that proportion. One part then moves to each of the two adjacent sites. This paper considers a continuous analogue of this evolution, which may be described by means of a stochastic flow of kernels, the theory of which was developed by Le Jan and Raimond. One of their results is that such a flow is characterized by specifying its N point motions, which form a consistent family of Brownian motions. This means for each dimension N we have a diffusion in R^N, whose N coordinates are all Brownian motions. Any M coordinates taken from the N-dimensional process are distributed as the M-dimensional process in the family. Moreover, in this setting, the only interactions between coordinates are local: when coordinates differ in value they evolve independently of each other. In this paper we explain how such multidimensional diffusions may be constructed and characterized via martingale problems.

We solve the largest sample eigenvalue distribution problem in the rank 1 spiked model of the quaternionic Wishart ensemble, which is the first case of a statistical generalization of the Laguerre symplectic ensemble (LSE) on the soft edge. We observe a phase change phenomenon similar to that in the complex case, and prove that the new distribution at the phase change point is the GOE Tracy–Widom distribution.

By bootstrap percolation we mean the following deterministic process on a graph G. Given a set A of vertices “infected” at time 0, new vertices are subsequently infected, at each time step, if they have at least r∈ℕ previously infected neighbors. When the set A is chosen at random, the main aim is to determine the critical probability p_c(G, r) at which percolation (infection of the entire graph) becomes likely to occur. This bootstrap process has been extensively studied on the d-dimensional grid [n]^d: with 2≤r≤d fixed, it was proved by Cerf and Cirillo (for d=r=3), and by Cerf and Manzo (in general), that $$p_{c}([n]^{d},r)=\Theta\biggl(\frac{1}{\log_{(r-1)}n}\biggr)^{d-r+1}$$ where log_(r) is an r-times iterated logarithm. However, the exact threshold function is only known in the case d=r=2, where it was shown by Holroyd to be $(1+o(1))\frac{\pi^{2}}{18\log n}$. In this paper we shall determine the exact threshold in the crucial case d=r=3, and lay the groundwork for solving the problem for all fixed d and r.

We develop some theory of spinal decompositions of discrete and continuous fragmentation trees. Specifically, we consider a coarse and a fine spinal integer partition derived from spinal tree decompositions. We prove that for a two-parameter Poisson–Dirichlet family of continuous fragmentation trees, including the stable trees of Duquesne and Le Gall, the fine partition is obtained from the coarse one by shattering each of its parts independently, according to the same law. As a second application of spinal decompositions, we prove that among the continuous fragmentation trees, stable trees are the only ones whose distribution is invariant under uniform re-rooting.

Fix ν>0, denote by G(ν/2) a Gamma random variable with parameter ν/2 and let n≥2 be a fixed even integer. Consider a sequence {F_k}_k≥1 of square integrable random variables belonging to the nth Wiener chaos of a given Gaussian process and with variance converging to 2ν. As k→∞, we prove that F_k converges in distribution to 2G(ν/2)−ν if and only if E(F_k⁴)−12E(F_k³)→12ν²−48ν.

We consider the stochastic reflection problem associated with a self-adjoint operator A and a cylindrical Wiener process on a convex set K with nonempty interior and regular boundary Σ in a Hilbert space H. We prove the existence and uniqueness of a smooth solution for the corresponding elliptic infinite-dimensional Kolmogorov equation with Neumann boundary condition on Σ.

We consider vector valued, unit variance Gaussian processes defined over stratified manifolds and the geometry of their excursion sets. In particular, we develop an explicit formula for the expectation of all the Lipschitz–Killing curvatures of these sets. Whereas our motivation is primarily probabilistic, with statistical applications in the background, this formula has also an interpretation as a version of the classic kinematic fundamental formula of integral geometry. All of these aspects are developed in the paper.

Particularly novel is the method of proof, which is based on a an approximation to the canonical Gaussian process on the n-sphere. The n→∞ limit, which gives the final result, is handled via recent extensions of the classic Poincaré limit theorem.

We study the threshold θ contact process on ℤ^d with infection parameter λ. We show that the critical point λ_c, defined as the threshold for survival starting from every site occupied, vanishes as d→∞. This implies that the threshold θ voter model on ℤ^d has a nondegenerate extremal invariant measure, when d is large.

We consider a (sub-)critical Galton–Watson process with neutral mutations (infinite alleles model), and decompose the entire population into clusters of individuals carrying the same allele. We specify the law of this allelic partition in terms of the distribution of the number of clone-children and the number of mutant-children of a typical individual. The approach combines an extension of Harris representation of Galton–Watson processes and a version of the ballot theorem. Some limit theorems related to the distribution of the allelic partition are also given.

Mathematical mean-field approaches play an important role in different fields of Physics and Chemistry, but have found in recent works also their application in Economics, Finance and Game Theory. The objective of our paper is to investigate a special mean-field problem in a purely stochastic approach: for the solution (Y, Z) of a mean-field backward stochastic differential equation driven by a forward stochastic differential of McKean–Vlasov type with solution X we study a special approximation by the solution (X^N, Y^N, Z^N) of some decoupled forward–backward equation which coefficients are governed by N independent copies of (X^N, Y^N, Z^N). We show that the convergence speed of this approximation is of order $1/\sqrt{N}$. Moreover, our special choice of the approximation allows to characterize the limit behavior of $\sqrt{N}(X^{N}-X,Y^{N}-Y,Z^{N}-Z)$. We prove that this triplet converges in law to the solution of some forward–backward stochastic differential equation of mean-field type, which is not only governed by a Brownian motion but also by an independent Gaussian field.

We give a probabilistic proof of the Weyl integration formula on U(n), the unitary group with dimension n. This relies on a suitable definition of Haar measures conditioned to the existence of a stable subspace with any given dimension p. The developed method leads to the following result: for this conditional measure, writing Z_U^(p) for the first nonzero derivative of the characteristic polynomial at 1, $$\frac{Z_{U}^{(p)}}{p!}\stackrel{\mathrm{law}}{=}\prod_{\ell =1}^{n-p}(1-X_{\ell})$$ the X_ℓ’s being explicit independent random variables. This implies a central limit theorem for log Z_U^(p) and asymptotics for the density of Z_U^(p) near 0. Similar limit theorems are given for the orthogonal and symplectic groups, relying on results of Killip and Nenciu.

Let P_t be the diffusion semigroup generated by L:=Δ+∇V on a complete connected Riemannian manifold with Ric≥−(σ²ρ_o²+c) for some constants σ, c>0 and ρ_o the Riemannian distance to a fixed point. It is shown that P_t is hypercontractive, or the log-Sobolev inequality holds for the associated Dirichlet form, provided −Hess_V≥δ holds outside of a compact set for some constant $\delta >(1+\sqrt{2})\sigma \sqrt{d-1}$. This indicates, at least in finite dimensions, that Ric and −Hess_V play quite different roles for the log-Sobolev inequality to hold. The supercontractivity and the ultracontractivity are also studied.

Let p_n(y)=∑_kα̂_kϕ(y−k)+∑_l=0^j_n−1∑_kβ̂_lk2^l/2ψ(2^ly−k) be the linear wavelet density estimator, where ϕ, ψ are a father and a mother wavelet (with compact support), α̂_k, β̂_lk are the empirical wavelet coefficients based on an i.i.d. sample of random variables distributed according to a density p₀ on ℝ, and j_n∈ℤ, j_n↗∞. Several uniform limit theorems are proved: First, the almost sure rate of convergence of sup _y∈ℝ|p_n(y)−Ep_n(y)| is obtained, and a law of the logarithm for a suitably scaled version of this quantity is established. This implies that sup _y∈ℝ|p_n(y)−p₀(y)| attains the optimal almost sure rate of convergence for estimating p₀, if j_n is suitably chosen. Second, a uniform central limit theorem as well as strong invariance principles for the distribution function of p_n, that is, for the stochastic processes $\sqrt{n}(F_{n}^{W}(s)-F(s))=\sqrt{n}\int_{-\infty}^{s}(p_{n}-p_{0})$, s∈ℝ, are proved; and more generally, uniform central limit theorems for the processes $\sqrt{n}\int(p_{n}-p_{0})f$, $f\in\mathcal{F}$, for other Donsker classes $\mathcal{F}$ of interest are considered. As a statistical application, it is shown that essentially the same limit theorems can be obtained for the hard thresholding wavelet estimator introduced by Donoho et al. [Ann. Statist. 24 (1996) 508–539].