The goal of this work is to find the asymptotics of the hitting probability of a distant point for the voter model on the integer lattice started from a single 1 at the origin. In dimensions d=2 or 3, we obtain the precise asymptotic behavior of this probability. We use the scaling limit of the voter model started from a single 1 at the origin in terms of super-Brownian motion under its excursion measure. This invariance principle was stated by Bramson, Cox and Le Gall, as a consequence of a theorem of Cox, Durrett and Perkins. Less precise estimates are derived in dimension d≥4.

We consider the supercritical oriented percolation model. Let ${\mathscr {K}}$ be all the percolation points. For each $u\in{ \mathscr {K}}$, we write γ_u as its rightmost path. Let G=⋃_uγ_u. In this paper, we show that G is a single tree with only one topological end. We also present a relationship between ${\mathscr {K}}$ and G and construct a bijection between ${\mathscr {K}}$ and ℤ using the preorder traversal algorithm. Through applications of this fundamental graph property, we show the uniqueness of an infinite oriented cluster by ignoring finite vertices.

We study random vectors of the form (Tr(A⁽¹⁾V), …, Tr(A^(r)V)), where V is a uniformly distributed element of a matrix version of a classical compact symmetric space, and the A^(ν) are deterministic parameter matrices. We show that for increasing matrix sizes these random vectors converge to a joint Gaussian limit, and compute its covariances. This generalizes previous work of Diaconis et al. for Haar distributed matrices from the classical compact groups. The proof uses integration formulas, due to Collins and Śniady, for polynomial functions on the classical compact groups.

We consider weakly damped nonlinear Schrödinger equations perturbed by a noise of small amplitude. The small noise is either complex and of additive type or real and of multiplicative type. It is white in time and colored in space. Zero is an asymptotically stable equilibrium point of the deterministic equations. We study the exit from a neighborhood of zero, invariant under the flow of the deterministic equation, in L² or in H¹. Due to noise, large fluctuations from zero occur. Thus, on a sufficiently large time scale, exit from these domains of attraction occur. A formal characterization of the small noise asymptotic of both the first exit times and the exit points is given.

Using time-reversal, we introduce a stochastic integral for zero-energy additive functionals of symmetric Markov processes, extending earlier work of S. Nakao. Various properties of such stochastic integrals are discussed and an Itô formula for Dirichlet processes is obtained.

We prove that the martingale dimensions for canonical diffusion processes on a class of self-similar sets including nested fractals are always one. This provides an affirmative answer to the conjecture of S. Kusuoka [Publ. Res. Inst. Math. Sci. 25 (1989) 659–680].

Local limit theorems are derived for the number of occupied urns in general finite and infinite urn models under the minimum condition that the variance tends to infinity. Our results represent an optimal improvement over previous ones for normal approximation.

We give necessary and sufficient conditions for the (bounded) law of the iterated logarithm for canonical U-statistics of arbitrary order d, extending the previously known results for d=2. The nasc’s are expressed as growth conditions on a parameterized family of norms associated with the U-statistics kernel.

We prove that the Airy process, $\mathscr{A}(t)$, locally fluctuates like a Brownian motion. In the same spirit we also show that, in a certain scaling limit, the so-called discrete polynuclear growth process (PNG) behaves like a Brownian motion.

In the setting of multidimensional diffusions in random environment, we carry on the investigation of condition (T'), introduced by Sznitman [Ann. Probab. 29 (2001) 723–764] and by Schmitz [Ann. Inst. H. Poincaré Probab. Statist. 42 (2006) 683–714] respectively in the discrete and continuous setting, and which implies a law of large numbers with nonvanishing limiting velocity (ballistic behavior) as well as a central limit theorem. Specifically, we show that when d≥2, (T') is equivalent to an effective condition that can be checked by local inspection of the environment. When d=1, we prove that condition (T') is merely equivalent to almost sure transience. As an application of the effective criterion, we show that when d≥4 a perturbation of Brownian motion by a random drift of size at most ɛ>0 whose projection on some direction has expectation bigger than ɛ^2−η, η>0, satisfies condition (T') when ɛ is small and hence exhibits ballistic behavior. This class of diffusions contains new examples of ballistic behavior which in particular do not fulfill the condition in [Ann. Inst. H. Poincaré Probab. Statist. 42 (2006) 683–714], (5.4) therein, related to Kalikow’s condition.

We study asymptotic properties of the Green metric associated with transient random walks on countable groups. We prove that the rate of escape of the random walk computed in the Green metric equals its asymptotic entropy. The proof relies on integral representations of both quantities with the extended Martin kernel. In the case of finitely generated groups, where this result is known (Benjamini and Peres [Probab. Theory Related Fields 98 (1994) 91–112]), we give an alternative proof relying on a version of the so-called fundamental inequality (relating the rate of escape, the entropy and the logarithmic volume growth) extended to random walks with unbounded support.

The (standard) Brownian web is a collection of coalescing one- dimensional Brownian motions, starting from each point in space and time. It arises as the diffusive scaling limit of a collection of coalescing random walks. We show that it is possible to obtain a nontrivial limiting object if the random walks in addition branch with a small probability. We call the limiting object the Brownian net, and study some of its elementary properties.