We consider stochastic growth models, such as standard first-passage percolation on $\mathbb{Z}^d$, where to leading order there is a linearly growing deterministic shape. Under natural hypotheses, we prove that for $d = 2$, the shape fluctuations grow at least logarithmically in all directions. Although this bound is far from the expected power law behavior with exponent $\chi = 1/3$, it does prove divergence. With additional hypotheses, we obtain inequalities involving $\chi$ and the related exponent $\xi$ (which is expected to equal 2/3 for $d = 2$). Combining these inequalities with previously known results, we obtain for standard first-passage percolation the bounds $\chi \geq 1/8$ for $d = 2$ and $\xi \leq 3/4$ for all $d$.

We consider a Brownian motion in a Poissonian potential conditioned to reach a remote location. We show that for typical configurations the expectation of the time $H$ to reach this goal grows at most linearly in the distance from the goal to the origin. In spite of the fact that $H$ has no finite exponential moment, we derive three exponential estimates, one of which concerns the size of a natural lattice animal attached to the trajectory of the process up to the goal.

Let $\mathscr{G}$ be a (possibly directed) locally finite graph with countably infinite vertex set $\mathfrak{V}$. Let $\{X(v): v \in \mathfrak{V}\}$ be an i.i.d. family of random variables with $P\{X(v) = 1\} = 1 - P\{X(v) = 0\} = p$. Finally, let $\xi = (\xi_1, \xi_2,\ldots)$ be a generic element of $\{0, 1\}^{\mathbb{N}}$; such a $\xi$ is called a word. We say that the word $\xi$ is seen from the vertex $v$ if there exists a self-avoiding path $(v, v_1, v_2, \ldots)$ on $\mathscr{G}$ starting at $v$ and such that $X(v_i) = \xi_i$ for $i \geq 1$. The traditional problem in (site) percolation is whether $P\{(1, 1, 1, \ldots)$ is seen from $v\} > 0$. So-called $AB$-percolation occurs if $P\{(1, 0, 1, 0, 1, 0,\ldots)$ is seen from $v\} > 0$. Here we investigate (a) whether $P\{$all words are seen from $v\} = 1$. We show that both answers are positive if $\mathscr{G} = \mathbb{Z}^d$, or even $\mathbb{Z}^d_+$ with all edges oriented in the "positive direction," when $d$ is sufficiently large. We show that on the oriented $\mathbb{Z}^3_+$ the answer to (a) is negative, but we do not know the answer to (b) on $\mathbb{Z}^3_+$. Various graphs $\mathscr{G}$ are constructed (almost all of them trees) for which the set of words $\xi$ which can be seen from a given $v$ (or from some $v$) is large, even though it is w.p.1 not the set of all words.

We consider a continuous time simple random walk moving among obstacles, which are sites (resp., bonds) of the lattice $Z^d$. We derive in this context a version of the technique of enlargement of obstacles developed by Sznitman in the Brownian case. This method gives controls on exponential moments of certain death times as well as good lower bounds for certain principal eigenvalues. We give an application to recover an asymptotic result of Donsker and Varadhan on the number of sites visited by the random walk and another application to the number of bonds visited by the random walk.

Evans defined a notion of what it means for a set $B$ to be polar for a process indexed by a tree. The main result herein is that a tree picked from a Galton-Watson measure whose offspring distribution has mean $m$ and finite variance will almost surely have precisely the same polar sets as a deterministic tree of the same growth rate. This implies that deterministic and nondeterministic trees behave identically in a variety of probability models. Mapping subsets of Euclidean space to trees and polar sets to capacity criteria, it follows that certain random Cantor sets are capacity-equivalent to each other and to deterministic Cantor sets. An extension to branching processes in varying environment is also obtained.

The Kesten-Stigum theorem is a fundamental criterion for the rate of growth of a supercritical branching process, showing that an $L \log L$ condition is decisive. In critical and subcritical cases, results of Kolmogorov and later authors give the rate of decay of the probability that the process survives at least $n$ generations. We give conceptual proofs of these theorems based on comparisons of Galton-Watson measure to another measure on the space of trees. This approach also explains Yaglom's exponential limit law for conditioned critical branching processes via a simple characterization of the exponential distribution.

A so-called generalized epidemic model is considered that describes the spread of an infectious disease of the SIR type with any specified distribution for the infectious period. The statistic under study is the number of susceptibles who ultimately survive the disease. In a pioneering paper, Daniels established for a particular case that when the population is large, this variable may have a Poisson-like behavior. This result was discussed later by several authors. In the present work, a necessary and sufficient condition is derived that guarantees the validity of such a Poisson approximation for the generalized epidemic. The proof relies on two key ideas, namely, the building of an equivalent Markovian representation of the model and the use of a suitable coupling via a random walk.

A connection between structural studies of stationary non-Gaussian stable processes and the ergodic theory of nonsingular flows is established and exploited. Using this connection, a unique decomposition of a stationary stable process into three independent stationary parts is obtained. It is shown that the dissipative part of a flow generates a mixed moving average part of a stationary stable process, while the identity part of a flow essentially gives the harmonizable part. The third part of a stationary process is determined by a conservative flow without fixed points and by a related cocycle.

Let $(X_i)_{i\in\mathbb{Z}}$ be a strictly stationary and strongly mixing sequence of real-valued mean zero random variables. Let $(\alpha_n)_{n > 0}$ be the sequence of strong mixing coefficients. We define the strong mixing function $\alpha(\cdot)$ by $\alpha(t) = \alpha_{\lbrack t\rbrack}$ and we denote by $Q$ the quantile function of $|X_0|$. Assume that \begin{equation*}\tag{*}\int^1_0\alpha^{-1}(t)Q^2(t) dt < \infty,\end{equation*} where $f^{-1}$ denotes the inverse of the monotonic function $f$. The main result of this paper is that the functional law of the iterated logarithm (LIL) holds whenever $(X_i)_{i\in\mathbb{Z}}$ satisfies $(\ast)$. Moreover, it follows from Doukhan, Massart and Rio that for any positive $a$ there exists a stationary sequence $(X_i)_{i\in\mathbb{Z}}$ with strong mixing coefficients $\alpha_n$ of the order of $n^{-a}$ such that the bounded LIL does not hold if condition $(\ast)$ is violated. The proof of the functional LIL is mainly based on new maximal exponential inequalities for strongly mixing processes, which are of independent interest.

Let $\{X_n\}$ be, for example, a weakly stationary sequence or a lacunary system with finite $p$th moment, $1 \leq p \leq 2$, and let $\{a_n\}$ be a sequence of scalars. We obtain here conditions which ensure the almost sure convergence of the series $\sum a_nX_n$. When $\{X_n\}$ is an orthonormal sequence, the classical Rademacher-Menchov theorem is recovered. This is then applied to study the strong consistency of least squares estimates in multiple regression models.

We prove a central limit theorem for the distance of the Brownian point on the universal cover of a compact negatively curved Riemannian manifold. The technical point is a contraction property for the leafwise Brownian motion along the stable foliation.

Let $V_W$ and $V_Y$ be Euclidean vector spaces and let $V_Z \equiv L(V_W \rightarrow V_Y)$. Given a Wiener process $W$ on $V_W$, with natural filtration $\{\mathscr{I}_t\}$, and a $\mathscr{I}_T$-measurable random variable $U$ in $V_Y$, we seek adapted processes $(Y, Z)$ in $V_Y \times V_Z$ satisfying the SDE $U = Y(t) + \int_{(t,T\rbrack}ZdW - \int_{(t, T\rbrack} \Gamma(Y, ZZ^\ast) ds/2, 0 \leq t \leq T,$ under local Lipschitz and convexity conditions on the map $(y, A) \rightarrow \Gamma(y, A)$. These conditions apply in particular in the case $\Gamma(y, A) = \sum\Gamma^i_{jk}(y)A^{jk}$, where $\Gamma$ is a linear connection on $V_Y$ whose Christoffel symbols $\Gamma^i_{jk}$ are bounded and Lipschitz, and $\Gamma$ has certain convexity properties. In that case the solution $Y$ above is known as a $\Gamma$-martingale with terminal value $U$. The solution $(Y, Z)$ is constructed explicitly using the Pardoux-Peng theory of backwards SDE's. Applications include the Dirichlet problem and the heat equation for harmonic mappings, and other PDE's.

The purpose of this paper is to prove a characterization of the conditional independence of two independent random variables given a particular functional of them, in terms of a factorization property. As an application we discuss the Markov field property for solutions of stochastic differential equations with a boundary condition involving the values of the process at times $t = 0$ and $t = 1$.

Let $B$ be a brownian motion starting at 0. We denote by $L^\ast_t = \max_{x\in\mathbb{R}} L^\ast_t$ the maximum of local times at time $t$. The Barlow-Yor inequalities tell us that for every $p > 0$, there are constants $C_p > c_p > 0$ such that for every stopping time $\tau$, $c_p\mathbb{E}\lbrack\tau^{p/2}\rbrack \leq \mathbb{E}\lbrack L^{\ast p}_\tau\rbrack \leq C_p\mathbb{E}\lbrack\tau^{p/2}\rbrack.$ Given a fixed closed set $F \subset \mathbb{R}$, we give a condition on $F$ which is necessary and sufficient to derive similar inequalities with $\max_{x\in F}L^x_\tau$ instead of $L^\ast_\tau$ and we prove various related results.

If $(X_t)$ is a one-dimensional diffusion corresponding to the operator $\mathscr{L} = \frac{1}{2}\partial_{xx} - \alpha\partial_x$ starting from $x > 0$ and $T_a$ is the hitting time of $a$, we prove that under suitable conditions on the drift coefficient the following limit exists: $\forall s > 0, \forall A \in \mathscr{F}_s, \lim_{t\rightarrow\infty} \mathbb{P}_x(X \in A\mid T_0 > t)$. We characterize this limit as the distribution of an $h$-like process, $h$ satisfying $\mathscr{L}h = - \eta h, h(0) = 0, h'(0) = 1$, where $\eta = -\lim_{t\rightarrow\infty}(1/t)\log\mathbb{P}_x(T_0 > t)$. Moreover, we show that this parameter $\eta$ can only take two values: $\eta = 0$ or $\eta = \underline{\lambda}$, where $\underline{\lambda}$ is the smallest point of increase of the spectral distribution of the operator $\mathscr{L}^\ast = \frac{1}{2}\partial_{xx} + \partial_x(\alpha\cdot)$.

For a system of infinitely many independent symmetric random walks on $\mathbb{Z}$ let $K_n(x)$ be the number of visits to $x \in \mathbb{Z}$ from time 0 to $n - 1$. The probabilities of some rare events involving $(K_n(0), K_n(1))$ are estimated as $n \rightarrow \infty$ and the corresponding large deviation rate functions are derived for both deterministic and invariant initial distributions. The dependence on the initial distributions is discussed. A simple method is used for guessing at the rate functions. This method is effective for independent random walks on the line and is worth exploring in more general settings.

The probability that a transient Markov chain, or a Brownian path, will ever visit a given set $\Lambda$ is classically estimated using the capacity of $\Lambda$ with respect to the Green kernel $G(x, y)$. We show that replacing the Green kernel by the Martin kernel $G(x, y)/G(0, y)$ yields improved estimates, which are exact up to a factor of 2. These estimates are applied to random walks on lattices and also to explain a connection found by Lyons between capacity and percolation on trees.

Assemblies are the decomposable combinatorial constructions characterized by the exponential formula for generating functions: $\Sigma p(n)s^n/n! = \exp(\Sigma m_is^i/i!)$. Here $p(n)$ is the total number of constructions that can be formed from a set of size $n$, and $m_n$ is the number of these structures consisting of a single component. Examples of assemblies include permutations, graphs, 2-regular graphs, forests of rooted or unrooted trees, set partitions and mappings of a set into itself. If an assembly is chosen uniformly from all possibilities on a set of size $n$, the counts $C_i(n)$ of components of size $i$ are jointly distributed like independent nonidentically distributed Poisson variables $Z_i$ conditioned on the event $Z_1 + 2Z_2 + \cdots + nZ_n = n$. We consider assemblies for which the process of component-size counts has a nontrivial limit distribution, without renormalizing. These include permutations, mappings, forests of labelled trees and 2-regular graphs, but not graphs and not set partitions. For some of these assemblies, the distribution of the component sizes may be viewed as a perturbation of the Ewens sampling formula with parameter $\theta$. We consider $d_b(n)$, the total variation distance between $(Z_1, \ldots, Z_b)$ and $(C_1(n),\ldots,C_b(n))$, counting components of size at most $b$. If the generating function of an assembly satisfies a mild analytic condition, we can determine the decay rate of $d_b(n)$. In particular, for $b = b(n) = o(n/\log n)$ and $n \rightarrow \infty, d_b(n) = o(b/n)$ if $\theta = 1$ and $d_b(n) \sim c(b)b/n$ if $\theta \neq 1$. The constant $c(b)$ is given explicitly in terms of the $m_i: c(b) = |1 - \theta|\mathbb{E}|T_{0b} - \mathbb{E}T_{0b}|/(2b)$, where $T_{0b} = Z_1 + 2Z_2 + \cdots + bZ_b$. Finally, we show that for $\theta \neq 1$ there is a constant $c_\theta$ such that $c(b) \sim c_\theta b$ as $b \rightarrow \infty$. Our results are proved using coupling, large deviation bounds and singularity analysis of generating functions.

We consider conditional propagation of chaos and use it to solve a class of quasilinear equations of parabolic type. In addition, we construct a class of continuous stochastic processes associated with the above nonlinear equations. Our method imposes fewer smoothness conditions on the coefficients and allows a degenerate nonlinear weight before a divergence form operator. We hope this probabilistic approach will introduce a better microscopic picture for understanding some Stefan type problems.

A principle of large deviations from the McKean-Vlasov limit is derived for a class of measure-valued jump processes. It is shown that the associated rate function admits several representations, including the one obtained by entropy methods and the one derived by a pathwise approach.

We consider infinite particle systems on a countable set $S$ with two-particle exclusion-eating motion determined by a symmetric transition function $p(x, y)$. This is, in a certain sense, a mixture of the exclusion process and the voter model. We discuss the dual process of this process and use the dual process to give a description of the set of invariant measures and to prove an ergodic theorem.