The Brownian web (BW) is the random network formally consisting of the paths of coalescing one-dimensional Brownian motions starting from every space-time point in ℝ×ℝ. We extend the earlier work of Arratia and of Tóth and Werner by providing a new characterization which is then used to obtain convergence results for the BW distribution, including convergence of the system of all coalescing random walks to the BW under diffusive space-time scaling.

For noncorrelated random variables, we study a concentration property of the family of distributions of normalized sums formed by sequences of times of a given large length.

We consider directed first-passage and last-passage percolation on the nonnegative lattice ℤ₊^d, d≥2, with i.i.d. weights at the vertices. Under certain moment conditions on the common distribution of the weights, the limits g(x)=lim _n→∞n⁻¹T(⌊nx⌋) exist and are constant a.s. for x∈ℝ₊^d, where T(z) is the passage time from the origin to the vertex z∈ℤ₊^d. We show that this shape function g is continuous on ℝ₊^d, in particular at the boundaries. In two dimensions, we give more precise asymptotics for the behavior of g near the boundaries; these asymptotics depend on the common weight distribution only through its mean and variance. In addition we discuss growth models which are naturally associated to the percolation processes, giving a shape theorem and illustrating various possible types of behavior with output from simulations.

We consider the (unoriented) long-range percolation on ℤ^d in dimensions d≥1, where distinct sites x,y∈ℤ^d get connected with probability p_xy∈[0,1]. Assuming p_xy=|x−y|^−s+o(1) as |x−y|→∞, where s>0 and |⋅| is a norm distance on ℤ^d, and supposing that the resulting random graph contains an infinite connected component C_∞, we let D(x,y) be the graph distance between x and y measured on C_∞. Our main result is that, for s∈(d,2d), D(x,y)=(log|x−y|)^Δ+o(1), x,y∈C_∞, |x−y|→∞, where Δ⁻¹ is the binary logarithm of 2d/s and o(1) is a quantity tending to zero in probability as |x−y|→∞. Besides its interest for general percolation theory, this result sheds some light on a question that has recently surfaced in the context of “small-world” phenomena. As part of the proof we also establish tight bounds on the probability that the largest connected component in a finite box contains a positive fraction of all sites in the box.

Benjamini, Lyons and Schramm [Random Walks and Discrete Potential Theory (1999) 56–84] considered properties of an infinite graph G, and the simple random walk on it, that are preserved by random perturbations. In this paper we solve several problems raised by those authors. The anchored expansion constant is a variant of the Cheeger constant; its positivity implies positive lower speed for the simple random walk, as shown by Virág [Geom. Funct. Anal. 10 (2000) 1588–1605]. We prove that if G has a positive anchored expansion constant, then so does every infinite cluster of independent percolation with parameter p sufficiently close to 1; a better estimate for the parameters p where this holds is in the Appendix. We also show that positivity of the anchored expansion constant is preserved under a random stretch if and only if the stretching law has an exponential tail. We then study a simple random walk in the infinite percolation cluster in Cayley graphs of certain amenable groups known as “lamplighter groups.” We prove that zero speed for a random walk on a lamplighter group implies zero speed for random walk on an infinite cluster, for any supercritical percolation parameter p. For p large enough, we also establish the converse.

We consider random walks in a random environment of the type p₀+γξ_z, where p₀ denotes the transition probabilities of a stationary random walk on ℤ^d, to nearest neighbors, and ξ_z is an i.i.d. random perturbation. We give an explicit expansion, for small γ, of the asymptotic speed of the random walk under the annealed law, up to order 2. As an application, we construct, in dimension d≥2, a walk which goes faster than the stationary walk under the mean environment.

We obtain Gaussian upper and lower bounds on the transition density q_t(x,y) of the continuous time simple random walk on a supercritical percolation cluster ${\mathcal{C}}_{\infty}$ in the Euclidean lattice. The bounds, analogous to Aronsen’s bounds for uniformly elliptic divergence form diffusions, hold with constants c_i depending only on p (the percolation probability) and d. The irregular nature of the medium means that the bound for q_t(x,⋅) holds only for t≥S_x(ω), where the constant S_x(ω) depends on the percolation configuration ω.

We consider a stochastic flow in which individual particles follow skew Brownian motions, with each one of these processes driven by the same Brownian motion. One does not have uniqueness for the solutions of the corresponding stochastic differential equation simultaneously for all real initial conditions. Due to this lack of the simultaneous strong uniqueness for the whole system of stochastic differential equations, the flow contains lenses, that is, pairs of skew Brownian motions which start at the same point, bifurcate, and then coalesce in a finite time. The paper contains qualitative and quantitative (distributional) results on the geometry of the flow and lenses.

We prove weak Poincaré inequalities on domains which are inverse images of open sets in Wiener spaces under continuous functions of Brownian rough paths. The result is applicable to Dirichlet forms on loop groups and connected open subsets of path spaces over compact Riemannian manifolds.

We extend the classical Douglas integral, which expresses the Dirichlet integral of a harmonic function on the unit disk in terms of its value on boundary, to the case of conservative symmetric diffusion in terms of Feller measure, by using the approach of time change of Markov processes.

For continuous γ, g:[0,1]→(0,∞), consider the degenerate stochastic differential equation $$dX_t=[1−|X_t|^2]^{1/2}γ(|X_t|) dB_t−g(|X_t|)X_t dt$$ in the closed unit ball of ℝⁿ. We introduce a new idea to show pathwise uniqueness holds when γ and g are Lipschitz and $\frac{g(1)}{\gamma^{2}(1)}>\sqrt{2}-1$. When specialized to a case studied by Swart [Stochastic Process. Appl. 98 (2002) 131–149] with $\gamma=\sqrt{2}$ and g≡c, this gives an improvement of his result. Our method applies to more general contexts as well. Let D be a bounded open set with C³ boundary and suppose $h: \overline D\to \mathbb {R}$ Lipschitz on $\overline D$, as well as C² on a neighborhood of ∂D with Lipschitz second partials there. Also assume h>0 on D, h=0 on ∂D and |∇h|>0 on ∂D. An example of such a function is h(x)=d(x,∂D). We give conditions which ensure pathwise uniqueness holds for $$dX_t=h(X_t)^{1/2}σ(X_t)dB_t+b(X_t) dt$$ in $\overline D$.

We present precise moderate deviation probabilities, in both quenched and annealed settings, for a recurrent diffusion process with a Brownian potential. Our method relies on fine tools in stochastic calculus, including Kotani’s lemma and Lamperti’s representation for exponential functionals. In particular, our result for quenched moderate deviations is in agreement with a recent theorem of Comets and Popov [Probab. Theory Related Fields 126 (2003) 571–609] who studied the corresponding problem for Sinai’s random walk in random environment.

If β_t is renormalized self-intersection local time for planar Brownian motion, we characterize when $\mathbb{E}e^{\gamma\beta_{1}}$ is finite or infinite in terms of the best constant of a Gagliardo–Nirenberg inequality. We prove large deviation estimates for β₁ and −β₁. We establish lim sup and lim inf laws of the iterated logarithm for β_t as t→∞.

Let α([0,1]^p) denote the intersection local time of p independent d-dimensional Brownian motions running up to the time 1. Under the conditions p(d−2)<d and d≥2, we prove $$\lim_{t\to\infty}t^{-1}\log \mathbb{P}\bigl\{\alpha([0,1]^{p})\ge t^{(d(p-1))/2}\bigr\}=-\gamma_{\alpha}(d,p)$$ with the right-hand side being identified in terms of the the best constant of the Gagliardo–Nirenberg inequality. Within the scale of moderate deviations, we also establish the precise tail asymptotics for the intersection local time $$I_n=\#\{(k_1,…,k_p)∈[1,n]^p ; S_1(k_1)=⋯=S_p(k_p)\}$$ run by the independent, symmetric, ℤ^d-valued random walks S₁(n), …,S_p(n). Our results apply to the law of the iterated logarithm. Our approach is based on Feynman–Kac type large deviation, time exponentiation, moment computation and some technologies along the lines of probability in Banach space. As an interesting coproduct, we obtain the inequality $$\bigl({\mathbb{E}}I_{n_{1}+\cdots +n_{a}}^{m}\bigr)^{1/p}\le \sum_{\mathop{k_{1}+\cdots +k_{a}=m}\limits_{k_{1},\ldots,k_{a}\ge 0}}{\frac{m!}{k_{1}!\cdots k_{a}!}}\bigl({\mathbb{E}}I_{n_{1}}^{k_{1}}\bigr)^{1/p}\cdots \bigl({\mathbb{E}}I_{n_{a}}^{k_{a}}\bigr)^{1/p}$$ in the case of random walks.

We consider the symmetric exclusion process {η_t,t>0} on {0,1}^ℤd. We fix a pattern ${\mathcal{A}}:=\{\eta: \sum_{\Lambda}\eta(i)\ge k\}$, where Λ is a finite subset of ℤ^d and k is an integer, and we consider the problem of establishing sharp estimates for τ, the hitting time of ${\mathcal{A}}$. We present a novel argument based on monotonicity which helps in some cases to obtain sharp tail asymptotics for τ in a simple way. Also, we characterize the trajectories {η_s,s≤t} conditioned on {τ>t}.

Let X₁,X₂,… be a sequence of [0,1]-valued i.i.d. random variables, let c≥0 be a sampling cost for each observation and let Y_i=X_i−ic, i=1,2,…. For n=1,2,…, let M(Y₁,…,Y_n)=E(max _1≤i≤nY_i) and V(Y₁,…,Y_n)=sup _τ∈CⁿE(Y_τ), where Cⁿ denotes the set of all stopping rules for Y₁,…,Y_n. Sharp upper bounds for the difference M(Y₁,…,Y_n)−V(Y₁,…,Y_n) are given under various restrictions on c and n.