In this paper we consider a one-dimensional diffusion equation on the interval [0, 1] satisfying non-Feller boundary conditions. As a consequence, the initial value Cauchy problem fails to preserve nonnegativity or boundedness. Nonetheless, probability theory plays an interesting role in our analysis and understanding of solutions to this equation.

Random planar maps are considered in the physics literature as the discrete counterpart of random surfaces. It is conjectured that properly rescaled random planar maps, when conditioned to have a large number of faces, should converge to a limiting surface whose law does not depend, up to scaling factors, on details of the class of maps that are sampled. Previous works on the topic, starting with Chassaing and Schaeffer, have shown that the radius of a random quadrangulation with n faces, that is, the maximal graph distance on such a quadrangulation to a fixed reference point, converges in distribution once rescaled by n^1/4 to the diameter of the Brownian snake, up to a scaling constant.

Using a bijection due to Bouttier, Di Francesco and Guitter between bipartite planar maps and a family of labeled trees, we show the corresponding invariance principle for a class of random maps that follow a Boltzmann distribution putting weight q_k on faces of degree 2k: the radius of such maps, conditioned to have n faces (or n vertices) and under a criticality assumption, converges in distribution once rescaled by n^1/4 to a scaled version of the diameter of the Brownian snake. Convergence results for the so-called profile of maps are also provided. The convergence of rescaled bipartite maps to the Brownian map, in the sense introduced by Marckert and Mokkadem, is also shown. The proofs of these results rely on a new invariance principle for two-type spatial Galton–Watson trees.

We consider a stochastic partial differential equation with reflection at 0 and with the constraint of conservation of the space average. The equation is driven by the derivative in space of a space–time white noise and contains a double Laplacian in the drift. Due to the lack of the maximum principle for the double Laplacian, the standard techniques based on the penalization method do not yield existence of a solution. We propose a method based on infinite dimensional integration by parts formulae, obtaining existence and uniqueness of a strong solution for all continuous nonnegative initial conditions and detailed information on the associated invariant measure and Dirichlet form.

The Skorokhod map is a convenient tool for constructing solutions to stochastic differential equations with reflecting boundary conditions. In this work, an explicit formula for the Skorokhod map Γ_0, a on [0, a] for any a>0 is derived. Specifically, it is shown that on the space $\mathcal{D}[0,\infty)$ of right-continuous functions with left limits taking values in ℝ, Γ_0, a=Λ_a○Γ₀, where $\Lambda_{a}: \mathcal{D}[0,\infty)\rightarrow\mathcal{D}[0,\infty )$ is defined by $$\Lambda_{a}(\phi)(t)=\phi(t)-\sup_{s\in[0,t]}\biggl[\bigl(\phi(s)-a\bigr)^{+}\wedge\inf_{u\in[s,t]}\phi(u)\biggr]$$ and $\Gamma_{0}: \mathcal{D}[0,\infty)\rightarrow\mathcal{D}[0,\infty)$ is the Skorokhod map on [0, ∞), which is given explicitly by $$\Gamma_{0}(\psi)(t)=\psi(t)+\sup_{s\in[0,t]}[-\psi(s)]^{+}.$$ In addition, properties of Λ_a are developed and comparison properties of Γ_0, a are established.

We prove a sufficient set of conditions for a sequence of finite measures on the space of cadlag measure-valued paths to converge to the canonical measure of super-Brownian motion in the sense of convergence of finite-dimensional distributions. The conditions are convergence of the Fourier transform of the r-point functions and perhaps convergence of the “survival probabilities.” These conditions have recently been shown to hold for a variety of statistical mechanical models, including critical oriented percolation, the critical contact process and lattice trees at criticality, all above their respective critical dimensions.

The goal is to show that an edge-reinforced random walk on a graph of bounded degree, with reinforcement weight function W taken from a general class of reciprocally summable reinforcement weight functions, traverses a random attracting edge at all large times.

The statement of the main theorem is very close to settling a conjecture of Sellke [Technical Report 94-26 (1994) Purdue Univ.]. An important corollary of this main result says that if W is reciprocally summable and nondecreasing, the attracting edge exists on any graph of bounded degree, with probability 1. Another corollary is the main theorem of Limic [Ann. Probab. 31 (2003) 1615–1654], where the class of weights was restricted to reciprocally summable powers.

The proof uses martingale and other techniques developed by the authors in separate studies of edge- and vertex-reinforced walks [Ann. Probab. 31 (2003) 1615–1654, Ann. Probab. 32 (2004) 2650–2701] and of nonconvergence properties of stochastic algorithms toward unstable equilibrium points of the associated deterministic dynamics [C. R. Acad. Sci. Sér. I Math. 330 (2000) 125–130].

We consider nonintersecting random walks satisfying the condition that the increments have a finite moment generating function. We prove that in a certain limiting regime where the number of walks and the number of time steps grow to infinity, several limiting distributions of the walks at the mid-time behave as the eigenvalues of random Hermitian matrices as the dimension of the matrices grows to infinity.

Coalescents with multiple collisions, also known as Λ-coalescents, were introduced by Pitman and Sagitov in 1999. These processes describe the evolution of particles that undergo stochastic coagulation in such a way that several blocks can merge at the same time to form a single block. In the case that the measure Λ is the Beta (2−α, α) distribution, they are also known to describe the genealogies of large populations where a single individual can produce a large number of offspring. Here, we use a recent result of Birkner et al. to prove that Beta-coalescents can be embedded in continuous stable random trees, about which much is known due to the recent progress of Duquesne and Le Gall. Our proof is based on a construction of the Donnelly–Kurtz lookdown process using continuous random trees, which is of independent interest. This produces a number of results concerning the small-time behavior of Beta-coalescents. Most notably, we recover an almost sure limit theorem of the present authors for the number of blocks at small times and give the multifractal spectrum corresponding to the emergence of blocks with atypical size. Also, we are able to find exact asymptotics for sampling formulae corresponding to the site frequency spectrum and the allele frequency spectrum associated with mutations in the context of population genetics.

The zero bias distribution W^* of W, defined though the characterizing equation EW f(W)=σ²E f'(W^*) for all smooth functions f, exists for all W with mean zero and finite variance σ². For W and W^* defined on the same probability space, the L¹ distance between F, the distribution function of W with EW=0 and Var(W)=1, and the cumulative standard normal Φ has the simple upper bound

‖F−Φ‖₁≤2E|W^*−W|.

This inequality is used to provide explicit L¹ bounds with moderate-sized constants for independent sums, projections of cone measure on the sphere S(ℓ_n^p), simple random sampling and combinatorial central limit theorems.

This paper derives sufficient conditions for superconvergence of sums of bounded free random variables and provides an estimate for the rate of superconvergence.

We develop a theory of ergodicity for a class of random dynamical systems where the driving noise is not white. The two main tools of our analysis are the strong Feller property and topological irreducibility, introduced in this work for a class of non-Markovian systems. They allow us to obtain a criteria for ergodicity which is similar in nature to the Doob–Khas’minskii theorem.

The second part of this article shows how it is possible to apply these results to the case of stochastic differential equations driven by fractional Brownian motion. It follows that under a nondegeneracy condition on the noise, such equations admit a unique adapted stationary solution.

We consider a recurrent random walk in a random environment on a regular tree. Under suitable general assumptions concerning the distribution of the environment, we show that the walk exhibits an unusually slow movement: the order of magnitude of the walk in the first n steps is (log n)³.

In this paper, we introduce some fundamental notions related to the so-called stochastic derivatives with respect to a given σ-field $\mathcal{Q}$. In our framework, we recall well-known results about Markov–Wiener diffusions. We then focus mainly on the case where X is a fractional diffusion and where $\mathcal{Q}$ is the past, the future or the present of X. We treat some crucial examples and our main result is the existence of stochastic derivatives with respect to the present of X when X solves a stochastic differential equation driven by a fractional Brownian motion with Hurst index H>1/2. We give explicit formulas.