This paper is based on Wald Lectures given at the annual meeting of the IMS in Minneapolis during August 2005. It is a survey of the theory of large deviations.

We consider invasion percolation on a rooted regular tree. For the infinite cluster invaded from the root, we identify the scaling behavior of its r-point function for any r≥2 and of its volume both at a given height and below a given height. We find that while the power laws of the scaling are the same as for the incipient infinite cluster for ordinary percolation, the scaling functions differ. Thus, somewhat surprisingly, the two clusters behave differently; in fact, we prove that their laws are mutually singular. In addition, we derive scaling estimates for simple random walk on the cluster starting from the root. We show that the invasion percolation cluster is stochastically dominated by the incipient infinite cluster. Far above the root, the two clusters have the same law locally, but not globally.

A key ingredient in the proofs is an analysis of the forward maximal weights along the backbone of the invasion percolation cluster. These weights decay toward the critical value for ordinary percolation, but only slowly, and this slow decay causes the scaling behavior to differ from that of the incipient infinite cluster.

We define a family of stochastic Loewner evolution-type processes in finitely connected domains, which are called continuous LERW (loop-erased random walk). A continuous LERW describes a random curve in a finitely connected domain that starts from a prime end and ends at a certain target set, which could be an interior point, or a prime end, or a side arc. It is defined using the usual chordal Loewner equation with the driving function being $\sqrt{2}B(t)$ plus a drift term. The distributions of continuous LERW are conformally invariant. A continuous LERW preserves a family of local martingales, which are composed of generalized Poisson kernels, normalized by their behaviors near the target set. These local martingales resemble the discrete martingales preserved by the corresponding LERW on the discrete approximation of the domain. For all kinds of targets, if the domain satisfies certain boundary conditions, we use these martingales to prove that when the mesh of the discrete approximation is small enough, the continuous LERW and the corresponding discrete LERW can be coupled together, such that after suitable reparametrization, with probability close to 1, the two curves are uniformly close to each other.

We consider nearest-neighbor self-avoiding walk, bond percolation, lattice trees, and bond lattice animals on ℤ^d. The two-point functions of these models are respectively the generating function for self-avoiding walks from the origin to x∈ℤ^d, the probability of a connection from the origin to x, and the generating functions for lattice trees or lattice animals containing the origin and x. Using the lace expansion, we prove that the two-point function at the critical point is asymptotic to const.|x|^2−d as |x|→∞, for d≥5 for self-avoiding walk, for d≥19 for percolation, and for sufficiently large d for lattice trees and animals. These results are complementary to those of [Ann. Probab. 31 (2003) 349–408], where spread-out models were considered. In the course of the proof, we also provide a sufficient (and rather sharp if d>4) condition under which the two-point function of a random walk on ℤ^d is asymptotic to const.|x|^2−d as |x|→∞.

Let X={X(t), t∈R₊} be a real-valued symmetric Lévy process with continuous local times {L_t^x, (t, x)∈R₊×R} and characteristic function Ee^iλX(t)=e^−tψ(λ). Let $$\sigma_{0}^{2}(x-y)=\frac{4}{\pi}\int^{\infty}_{0}\frac{\sin^{2}({\lambda(x-y)}/{2})}{{\psi(\lambda)}}\,d\lambda.$$ If σ₀²(h) is concave, and satisfies some additional very weak regularity conditions, then for any p≥1, and all t∈R₊, $$\lim_{h\downarrow0}\int_{a}^{b}\biggl|{\frac{L^{x+h}_{t}-L^{x}_{t}}{\sigma_{0}(h)}}\biggr|^{p}\,dx=2^{p/2}E|\eta|^{p}\int_{a}^{b}|L^{x}_{t}|^{p/2}\,dx$$ for all a, b in the extended real line almost surely, and also in L^m, m≥1. (Here η is a normal random variable with mean zero and variance one.)

This result is obtained via the Eisenbaum Isomorphism Theorem and depends on the related result for Gaussian processes with stationary increments, {G(x), x∈R¹}, for which E(G(x)−G(y))²=σ₀²(x−y); $$\lim_{h\to0}\int_{a}^{b}\biggl|\frac{G(x+h)-G(x)}{\sigma_{0}(h)}\biggr|^{p}\,dx=E|\eta|^{p}(b-a)$$ for all a, b∈R¹, almost surely.

This paper is a continuation of our previous research on quadratic harnesses, that is, processes with linear regressions and quadratic conditional variances. Our main result is a construction of a Markov process from given orthogonal and martingale polynomials. The construction uses a two-parameter extension of the Al-Salam–Chihara polynomials and a relation between these polynomials for different values of parameters.

We are concerned with a new type of supermartingale decomposition in the Max-Plus algebra, which essentially consists in expressing any supermartingale of class $(\mathcal{D})$ as a conditional expectation of some running supremum process. As an application, we show how the Max-Plus supermartingale decomposition allows, in particular, to solve the American optimal stopping problem without having to compute the option price. Some illustrative examples based on one-dimensional diffusion processes are then provided. Another interesting application concerns the portfolio insurance. Hence, based on the “Max-Plus martingale,” we solve in the paper an optimization problem whose aim is to find the best martingale dominating a given floor process (on every intermediate date), w.r.t. the convex order on terminal values.

In this paper we investigate three discrete or semi-discrete approximation schemes for reflected Brownian motion on bounded Euclidean domains. For a class of bounded domains D in ℝⁿ that includes all bounded Lipschitz domains and the von Koch snowflake domain, we show that the laws of both discrete and continuous time simple random walks on D∩2^−kℤⁿ moving at the rate 2^−2k with stationary initial distribution converge weakly in the space D([0, 1], ℝⁿ), equipped with the Skorokhod topology, to the law of the stationary reflected Brownian motion on D. We further show that the following “myopic conditioning” algorithm generates, in the limit, a reflected Brownian motion on any bounded domain D. For every integer k≥1, let {X^k_j2^−k, j=0, 1, 2, …} be a discrete time Markov chain with one-step transition probabilities being the same as those for the Brownian motion in D conditioned not to exit D before time 2^−k. We prove that the laws of X^k converge to that of the reflected Brownian motion on D. These approximation schemes give not only new ways of constructing reflected Brownian motion but also implementable algorithms to simulate reflected Brownian motion.

We show that random walk in uniformly elliptic i.i.d. environment in dimension ≥5 has at most one non zero limiting velocity. In particular this proves a law of large numbers in the distributionally symmetric case and establishes connections between different conjectures.

We consider two dynamical variants of Dvoretzky’s classical problem of random interval coverings of the unit circle, the latter having been completely solved by L. Shepp. In the first model, the centers of the intervals perform independent Brownian motions and in the second model, the positions of the intervals are updated according to independent Poisson processes where an interval of length ℓ is updated at rate ℓ^−α where α≥0 is a parameter. For the model with Brownian motions, a special case of our results is that if the length of the nth interval is c/n, then there are times at which a fixed point is not covered if and only if c<2 and there are times at which the circle is not fully covered if and only if c<3. For the Poisson updating model, we obtain analogous results with c<α and c<α+1 instead. We also compute the Hausdorff dimension of the set of exceptional times for some of these questions.

Let X={X_t, t≥0} be a càdlàg Lévy process, centered, with moments of all orders. There are two families of orthogonal polynomials associated with X. On one hand, the Kailath–Segall formula gives the relationship between the iterated integrals and the variations of order n of X, and defines a family of polynomials P₁(x₁), P₂(x₁, x₂), … that are orthogonal with respect to the joint law of the variations of X. On the other hand, we can construct a sequence of orthogonal polynomials p_n^σ(x) with respect to the measure σ²δ₀(d x)+x²ν(d x), where σ² is the variance of the Gaussian part of X and ν its Lévy measure. These polynomials are the building blocks of a kind of chaotic representation of the square functionals of the Lévy process proved by Nualart and Schoutens. The main objective of this work is to study the probabilistic properties and the relationship of the two families of polynomials. In particular, the Lévy processes such that the associated polynomials P_n(x₁, …, x_n) depend on a fixed number of variables are characterized. Also, we give a sequence of Lévy processes that converge in the Skorohod topology to X, such that all variations and iterated integrals of the sequence converge to the variations and iterated integrals of X.

We consider functionals of long-range dependent Gaussian sequences with infinite variance and obtain nonstandard limit theorems. When the long-range dependence is strong enough, the limit is a Hermite process, while for weaker long-range dependence, the limit is α-stable Lévy motion. For the critical value of the long-range dependence parameter, the limit is a sum of a Hermite process and α-stable Lévy motion.