Majorizing measures provide bounds for the supremum of stochastic processes. They represent the most general possible form of the chaining argument going back to Kolmogorov. Majorizing measures arose from the theory of Gaussian processes, but they now have applications far beyond this setting. The fundamental question is the construction of these measures. This paper focuses on the tools that have been developed for this purpose and, in particular, the use of geometric ideas. Applications are given to several natural problems where entropy methods are powerless.

We obtain an integral formula for the density of the maximum of smooth Gaussian processes. This expression induces explicit nonasymptotic lower and upper bounds which are in general asymptotic to the density. Moreover, these bounds allow us to derive simple asymptotic formulas for the density with rate of approximation as well as accurate asymptotic bounds. In particular, in the case of stationary processes, the latter upper bound improves the well-known bound based on Rice's formula. In the case of processes with variance admitting a finite number of maxima, we refine recent results obtained by Konstant and Piterbarg in a broader context, producing the rate of approximation for suitable variants of their asymptotic formulas. Our constructive approach relies on a geometric representation of Gaussian processes involving a unit speed parameterized curve embedded in the unit sphere.

Let X be a strongly symmetric Hunt process with $\alpha$-potential density $u^\alpha(x,y). Let

$$ {\mathcal G}_{\alpha}^2 = \left\{\mu | \int\int(u^\alpha (x,y))^2 d\mu(x)\; d\mu (y)<\infty\right\}$$

and let $L_t^\mu$ denote the continuous additive functional with Revuz measure $\mu$. For a set of positive measures $M \subset G_\alpha^2$, subject to some additional regularity conditions, we consider families of continuous (in time) additive functionals $L = {L-t^\mu, (t, \mu) \in R^+ \times M} of X and a second-order Gaussian chaos $H_\alpha = {H_\alpha(\mu), \mu \in M}$ which is associated with L by an isomorphism theorem of Dynkin.

A general theorem is obtained which shows that, with some additional regularity conditions depending on X and M if $H_\alpha$ has a continuous version on M almost surely, then so does L and, furthermore, that moduli of continuity for $H_\alpha$ are also moduli of continuity for L.

Special attention is given to Lévy processes in $R^n$ and $T^n$, the n-dimensional torus, with $M$ taken to be the set of translates of a fixed measure. Many concrete examples are given, especially when X is Brownian motion in $R^n$ and $T^n$ for $n = 2$ and 3. For certain measures $\mu$ on $T^n$ and processes, including Brownian motion in $T^3$, necessary and sufficient conditions are given for the continuity of ${L_t^\mu, (t,\mu) \in R^+ \times M}$, where M is the set of all translates of $\mu$.

Let X be an exponentially killed Lévy process on $T^n$ , the $n$ -dimensional torus, that satisfies a sector condition. (This includes symmetric Lévy processes.) Let$\mathscr{F}_e$ denote the extended Dirichlet space of X. Let $h \subset \mathscr{F}_e$ and let ${h_y, y \ subset T^n}$ denote the set of translates of $h$. That is, $h_y(\dot) = h(\dot - y)$. We consider the family of zero-energy continuous additive functions ${N_t^{[h_y]}, (y,t) \subset T^n \times R^+}$ defined by Fukushima. For a very large class of random functions h we show that

$$J_\rho (T^n) = \int (\log N_\rho (T^n,\varepsilon))^{1/2} d\varepsilon < \infty$$

is a necessary and sufficient condition for the family ${N_t^{[h_y]}, (y,t) \subset T^n \times R^+}$ to have a continuous version almost surely. Here $N_p(T^n, \varepsilon)$ is the minimum number of balls of radius $\varepsilon$ in the metric p that covers $T^n$, where the metric p is the energy metric. We argue that this condition is the natural extension of the necessary and sufficient condition for continuity of local times of Lévy processes of Barlow and Hawkes.

Results on the bounded variation and p-variation (in t )of $N_t^{[h_y]}$, for y fixed, are also obtained for a large class of random functions h.

We study a wide class of transient planar graphs, through a geometric model given by a square tiling of a cylinder. For many graphs, the geometric boundary of the tiling is a circle and is easy to describe in general. The simple random walk on the graph converges (with probability 1) to a point in the geometric boundary. We obtain information on the harmonic measure and estimates on the rate of convergence. This allows us to extend results we previously proved for triangulations of a disk.

For a class of diffusions X in the plane we construct a global system of coordinates $u(x)$, such that $u(x)$ is close to X at infinity and $u(X)$ is a local martingale. Such coordinates are useful for the study of the long term behaviour of X. The construction uses probabilistic methods, in particular a coupling for general diffusions in the plane.

We derive the asymptotic laws of winding numbers for planar isotropic stable Lévy processes and walks of index $\alpha\subset (0,2)$

The planar Brownian snake is a continuous, strong Markov process taking values in the space of continuous functions in $\mathbb{R}^2$ that are stopped at some time. For a fixed time the snake is distributed like a planar Brownian motion with a random lifetime. This paper characterizes the convex hull of the trace of the snake paths that exit the half-plane at the origin. It is shown that the convex hull at 0 is roughly a factor of x smoother than the convex hull of a piece of planar Brownian motion at its minimum y-value.

We prove the strong law of large numbers for a continuous-time version of reinforced random walk. This extends previous results of Durrett and Rogers.

We consider non-Markovian, self-interacting random walks (SIRW) on the one-dimensional integer lattice. The walk starts from the origin and at each step jumps to a neighboring site. The probability of jumping along a bond is proportional to w (number of previous jumps along that lattice bond), where $w: \mathbb{N} \to \math{R}_+$ is a monotone weight function. Exponential and subexponential weight functions were considered in earlier papers. In the present paper we consider weight functions w with polynomial asymptotics. These weight functions define variants of the "reinforced random walk." We prove functional limit theorems for the local time processes of these random walks and local limit theorems for the position of the random walker at late times. A generalization of the Ray-Knight theory of local time arises.

Let ${X_n, 1 \leq n < \infty}$ be a sequence of independent identically distributed random variables in the domain of attraction of a stable law with index $0 < \alpha < 2$. The limit of $x_n^{-1}\log P{S_n/ \max |X_i| \geq x_n}$ is found when $x_n \to \infty$ and $\x_n/n \to 0$. The large deviation result is used to prove the law of the iterated logarithm for the self-normalized partial sums.

We establish very general one-sided results on the lim sup behavior of increments of suitably normalized partial sums of i.i.d. random variables. Our main results apply to arbitrary nondegenerate positive random variables which need not have any finite moments. As a corollary we can show that such results also hold for not necessarily positive random variables whose negative parts have finite moment-generating functions.

Consider the contact process on a homogeneous tree with degree $d \geq 3$. Denote by

$$\lambda_c = \inf\{\lambda : P(o \in \xi^o_t \text{i.o.}) > 0\}$$

the critical value of local survival probability, where o is the root of the tree. Pemantle and Durrett and Schinazi both conjectured that the complete convergence theorem should hold if $\lambda >\lambda_c$. Here we answer the conjecture affirmatively. Furthermore, we will show that

$$P(o \in \xi^o_t \text{i.o.}) = 0 \quad \text{at $\lambda_c}.$$

Therefore, the conclusion of the complete convergence theorem cannot hold at $\lambda_c$

We prove a large deviation principle for the finite-dimensional marginals of the Gibbs distribution of the macroscopic "overlap" parameters in the Hopfield model in the case where the number of random "patterns" M , as a function of the system size N, satisfies lim sup $M(N) /N =0$. In this case, the rate function is independent of the disorder for almost all realizations of the patterns.

A random variable X is N-divisible if it can be decomposed into a random sum of N i.i.d. components, where N is a random variable independent of the components; X is N-stable if the components are rescaled copies of X. These N-divisible and N-stable random variables arise in a variety of stochastic models, including thinned renewal processes and subordinated Lévy and stable processes. We consider a general theory of N-divisibility and N-stability in the case where $E(N) < \infty$, based on a representation of the probability generating function of N in terms of its limiting Laplace. Stieltjes transform $\mathscr{l}$ We analyze certain topological semigroups of such p.g.f.’s in detail, and on this basis we extend existing characterizations of N-divisible and N-stable laws in terms of $\mathscr{l}$ . We apply the results to the aforementioned stochastic models.

The scaled-sample problem asks the following question: given a distribution on a normed linear space E, when do there exist constants ${\gamma_n} such that $X^{(j)}/\gamma_n}_{j=1}^n$ converges as $n \to \infty$ (in the Hausdorff metric given by the norm) to a fixed set K? (Here ${X^{(j)}}$ are i.i.d. with the given distribution). The main result presented here relates the convergence of scaled samples to a large deviation principle for single observations, thereby achieving a dimension-free description of the problem.

We associate certain translation invariant random metrics on $\mathbb{R}^d$ to Brownian motion evolving in a truncated Poissonian potential. These metrics behave over large distances, in an appropriate sense, like certain deterministic norms (the so-called Lyapounov exponents). We prove here upper bounds on the size of fluctuations of the metrics around their mean. Under an additional assumption of rotational invariance, we also derive upper bounds on the difference between the mean of the metrics and the Lyapounov norms.

For controlled Markov processes taking values in a Polish space, control problems with ergodic cost, infinite-horizon discounted cost and finite-horizon cost are studied. Each is posed as a convex optimization problem wherein one tries to minimize a linear functional on a closed convex set of appropriately defined occupation measures for the problem. These are characterized as solutions of a linear equation asssociated with the problem. This characterization is used to establish the existence of optimal Markov controls. The dual convex optimization problem is also studied.

We address the problem of the convergence to equilibrium of a general class of point processes, containing, in particular, the nonlinear mutually exciting point processes, an extension of the linear Hawkes processes, and give general conditions guaranteeing the existence of a stationary version and the convergence to equilibrium of a nonstationary version, both in distribution and in variation. We also give a new proof of a result of Kerstan concerning point processes with bounded intensity and general nonlinear dynamics satisfying a Lipschitz condition.

Let $X, X_n, n \ge 1$, be a sequence of independent identically distributed random variables. We give necessary and sufficient conditions for the strong law of large numbers

n^{-k/p} \sum_{1\lei_1 \le i_2 <\dots < i_k \le n} X_{i_1}X_{i_2}\dots X_{i_k} \to 0\quad\text{a.s.}

for $k =2$ without regularity conditions on $X$, for $k \geq 3$ in three cases: (i) symmetric X, (ii) $P \{X \leq 0\} =1 and (iii) regularly varying $P\{|X|}> x\}$ as $x \to \infty$, without further conditions, and for general X and k under a condition on the growth of the truncated mean of X. Randomized, centered, squared and decoupled strong laws and general normalizing sequences are also considered.

Bolthausen established a bound of order $1/\sqrt{n}$ on the rate of convergence in the central limit theorem for martingale difference arrays having bounded conditional moments of order 4. In the present paper it is shown how much this moment condition can be relaxed while maintaining the same rate of convergence. An example shows that, unlike in the i.i.d. case, a moment condition of order 3 is not enough. Furthermore, exact rates of convergence are derived for moment conditions of order between 2 and 3.