Every two-dimensional incompressible flow follows the level lines of some scalar function $\psi$ on $\mathbb{R}^2$; transport properties of the flow depend in part on whether all level lines are bounded. We study the structure of the level lines when $\psi$ is a stationary random field. We show that under mild hypotheses there is only one possible alternative to bounded level lines: the "treelike" random fields, which, for some interval of values of a, have a unique unbounded level line at each level a, with this line "winding through every region of the plane." If the random field has the FKG property, then only bounded level lines are possible. For stationary $C^2$ Gaussian random fields with covariance function decaying to 0 at $\infty$, the treelike property is the only alternative to bounded level lines provided the density of the absolutely continuous part of the spectral measure decays at $\infty$ "slower than exponentially," and only bounded level lines are possible if the covariance function is nonnegative.

The contact process on $Z^d$ is known to have only two fundamental types of behavior: survival and extinction. Recently Pemantle discovered that the phase structure for the contact process on a tree can be more complex. There are three possible types of behavior: strong survival, weak survival and extinction. He proved that all three occur on homogeneous trees in which each vertex has $d + 1$ neighbors, provided that $d \geq 3$, but he left open the case $d = 2$. Since $d = 1$ corresponds to $Z^1$, in which weak survival does not occur, $d = 2$ is the boundary case. In this paper, we complete this picture, by showing that weak survival does occur on the binary tree for appropriate parameter values. In doing so, we extend and develop techniques for obtaining upper and lower bounds for the critical values associated with strong and weak survival of the contact process on more general graphs.

Let $\mathbb{T}_d$ be a homogeneous tree in which every vertex has $d$ neighbors. A new proof is given that the contact process on $\mathbb{T}_d$ exhibits two phase transitions when $d \geq 3$, a behavior which distinguishes it from the contact process on $\mathbb{Z}^n$. This is the first proof which does not involve calculation of bounds on critical values, and it is much shorter than the previous proof for the binary tree, $\mathbb{T}_3$. The method is extended to prove the existence of an intermediate phase for a more general class of trees with exponential growth and certain symmetry properties, for which no such result was previously known.

The relationship between species number and area is an old problem in biology. We propose here an interacting particle system--the multitype voter model with mutation--as a mathematical model to study this problem. We analyze the species area curves of this model as the mutation rate $\alpha$ tends to zero. We obtain two basic types of behavior depending on the size of the spatial region under consideration. If the region is a square with area $\alpha^{-r}, r > 1$, then, for small $\alpha$, the number of species is of order $\alpha^{1-r}(\log \alpha)^2$, whereas if $r < 1$, the number of species is bounded.

In the threshold growth model on an integer lattice, the occupied set grows according to a simple local rule: a site becomes occupied iff it sees at least a threshold number of already occupied sites in its prescribed neighborhood. In this paper, we analyze the behavior of two-dimensional threshold growth dynamics started from a sparse Bernoulli density of occupied sites. We explain how nucleation of rare centers, invariant shapes and interaction between growing droplets influence the first passage time in the supercritical case. We also briefly address scaling laws for the critical case.

The hydrodynamical limit of asymmetric simple exclusion processes is given by an inviscid Burgers equation and its next-order correction is given by the viscous Burgers equation. The diffusivity can be characterized by an abstract formulation in a Hilbert space with the inverse of the diffusivity characterized by a variational formula. Alternatively, it can be described by the Green-Kubo formula. We give arguments that these two formulations are equivalent. We also derive two other variational formulas, one for the inverse of the diffusivity and one for the diffusivity itself, characterizing diffusivity as a supremum and as an infimum. These two formulas also provide an analytic criterion for deciding whether the diffusivity as defined by the linear response theory is symmetric. Furthermore, we prove the continuity of the diffusivity and a few other relations concerning diffusivity and solutions of the Euler-Lagrange equations of these variational problems.

We consider the Hopfield model with n neurons and an increasing number $p = p(n)$ of randomly chosen patterns. Under the condition $(p^3 \log p)/n \to 0$, we prove for every fixed choice of overlap parameters a central limit theorem as $n \to \infty$, which holds for almost all realizations of the random patterns. In the special case where the temperature is above the critical one and there is no external magnetic field, the condition $(p^2 \log p)/n \to 0$ suffices. As in the case of a finite number of patterns, the central limit theorem requires a centering which depends on the random patterns.

We give easily verifiable conditions under which a functional central limit theorem holds for additive functionals of symmetric simple exclusion and symmetric zero-range processes. Also a reversible exclusion model with speed change is considered. Let $\eta (t)$ be the configuration of the process at time t and let $f(\eta)$ be a function on the state space. The question is: For which functions f does $\lambda^{-1/2} \int_0^{\lambda t} f(\eta(s)) ds$ converge to a Brownian motion? A general but often intractable answer is given by Kipnis and Varadhan. In this article we determine what conditions beyond a mean-zero condition on $f(\eta)$ are required for the diffusive limit above. Specifically, we characterize the $H^{-1}$ space in an applicable way.

Our method of proof relies primarily on a sharp estimate on the "spectral gap" of the process and weak regularity properties for the invariant measures.

We give a lower bound on the spectral gap for symmetric zero-range processes. Under some conditions on the rate function, we show that the gap shrinks as $n^{-2}$, independent of the density, for the dynamics localized on a cube of size $n^d$. We follow the method outlined by Lu and Yau, where a similar spectral gap is proved for Kawasaki dynamics.

For the minimal spanning tree on n independent uniform points in the d-dimensional unit cube, the proportionate number of points of degree k is known to converge to a limit $\alpha_{k,d}$ as $n \to \infty$. We show that $\alpha_{k,d}$ converges to a limit $\alpha_k$ as $d \to \infty$ for each k. The limit $\alpha_k$ arose in earlier work by Aldous, as the asymptotic proportionate number of vertices of degree k in the minimum-weight spanning tree on k vertices, when the edge weights are taken to be independent, identically distributed random variables. We give a graphical alternative to Aldous's characterization of the $\alpha_k$.

A stepping-stone model with site space a continuous, hierarchical group is constructed via duality with a system of (delayed) coalescing "stable" Lévy processes. This model can be understood as a continuum limit of discrete state-space, two-allele, genetics models with hierarchically structured resampling and migration. The existence of a process rescaling limit on suitably related large space and time scales is established and interpreted in terms of the dynamics of cluster formation. This paper was inspired by recent work of Klenke.

We study the limiting behavior of large branching particle systems undergoing random motion, whose branching mechanism is affected by a random environment. The weak convergence result is established for a sequence of such particle systems and the limiting process is characterized as the unique solution of a martingale problem. The proof of uniqueness of the solution for the martingale problem requires an extension of standard duality techniques.

It is proved that the potential kernel of a recurrent, aperiodic random walk on the integer lattice $\mathbb{Z}^2$ admits an asymptotic expansion of the form $$(2 \pi \sqrt{|Q|})^{-1} \ln Q(x_2, -x_1) + \const + |x|^{-1} U_1 (\omega^x) + |x|^{-2} U_2 (\omega^x) + \dots ,$$ where $|Q|$ and $Q(\theta)$ are, respectively, the determinant and the quadratic form of the covariance matrix of the increment X of the random walk, $\omega^x = x/|x|$ and the $U_k (\omega)$ are smooth functions of $\omega, |\omega| = 1$, provided k that all the moments of X are finite. Explicit forms of $U_1$ and $U_2$ are given in terms of the moments of X.

Kaimanovich and Vershik described certain finitely generated groups of exponential growth such that simple random walk on their Cayley graph escapes from the identity at a sublinear rate, or equivalently, all bounded harmonic functions on the Cayley graph are constant. Here we focus on a key example, called $G_1$ by Kaimanovich and Vershik, and show that inward-biased random walks on $G_1$ move outward faster than simple random walk. Indeed, they escape from the identity at a linear rate provided that the bias parameter is smaller than the growth rate of $G_1$. These walks can be viewed as random walks interacting with a dynamical environment on $\mathbb{Z}$. The proof uses potential theory to analyze a stationary environment as seen from the moving particle.

Let $\alpha$ and $\beta$ be real numbers and $f \in C_0 [0, \infty)$. We study the existence and uniqueness of solutions g of the equation $g(t) = f(t) + \alpha \sup_{0 \leq s \leq t} g(s) + \beta \inf_{0 \leq s \leq t} g(s)$. Carmona, Petit, Le Gall, and Yor have shown existence or nonexistence and uniqueness for some $\alpha, \beta$. We settle the remaining cases. We study the nearest neighbor walk on the integers, which behaves just like fair random walk unless one neighbor has been visited and the other has not, when it jumps to the unvisited neighbor with probability p. If $p < 2/3$, we show these processes, scaled, converge to the solution of the equation above for Brownian paths, with $\alpha = \beta = (2p - 1)/p$.

We establish existence and uniqueness results for adapted solutions of backward stochastic differential equations (BSDE's) with two reflecting barriers, generalizing the work of El Karoui, Kapoudjian, Pardoux, Peng and Quenez. Existence is proved first by solving a related pair of coupled optimal stopping problems, and then, under different conditions, via a penalization method. It is also shown that the solution coincides with the value of a certain Dynkin game, a stochastic game of optimal stopping. Moreover, the connection with the backward SDE enables us to provide a pathwise (deterministic) approach to the game.

Consider a locally compact second countable topological transformation group acting measurably on an arbitrary space. We show that the distributions of two random elements X and $X'$ in this space agree on invariant sets if and only if there is a random transformation $\Gamma$ such that $\Gamma X$ has the same distribution as $X'$. Applying this to random fields in d dimensions under site shifts, we show further that these equivalent claims are also equivalent to site-average total variation convergence. This convergence result extends to amenable groups.

Let ${Y_n}_{n \in \mathbb{Z}_+}$ be a sequence of random variables in $\mathbb{R}^d$ and let $A \subset \mathbb{R}^d$. Then $\mathbf{P}\{Y_n \in A \text{for some $n$}\}$ is the hitting probability of the set A by the sequence ${Y_n}$. We consider the asymptotic behavior, as $m \to \infty$, of $\mathbf{P}\{Y_n \in mA \text{some $n$}\} = \mathbf{P}{\text{hitting $mA$}$ whenever (1) the probability law of $Y_n/n$ satisfies the large deviation principle and (2) the central tendency of $Y_n/n$ is directed away from the given set A. For a particular function $\tilde{I}$, we show $m \to \infty$, of $\mathbf{P}\{Y_n \in mA \text{some $n$}\} \approx \exp (-m \tilde{I}(A))$.

By combining the Berkes-Philipp blocking technique and the Csörgö-Révész quantile transform methods, we find that partial sums of an associated sequence can be approximated almost surely by partial sums of another sequence with Gaussian marginals. A crucial fact is that this latter sequence is still associated with covariances roughly bounded by the covariances of the original sequence, and that one can approximate it by an iid Gaussian process using the Berkes-Philipp method. We require that the original sequence has finite $(2 + r)$th moments, $r > 0$, and a power decay rate of a coefficient $u(n)$ which describes the covariance structure of the sequence. Based on this result, we obtain a strong invariance principle for associated sequences if $u(n)$ exponentially decreases to 0.

In this paper we establish weak convergence theorems for weighted empirical processes of strong mixing, $\rho$-mixing and associated sequences. We apply these results to obtain weak convergence of integral functionals of empirical processes and of mean residual life processes in reliability theory. To carry out the proofs, we develop two Rosenthal-type inequalities for strong mixing and associated sequences.

Several classes of functions are shown to be Donsker by an argument based on partitioning the sample space. One example is the class of all nondecreasing functions $f: \mathbb{R} \to \mathbb{R}$ such that $0 \leq f \leq F$ for a given function F with $\int F^2 dP/ \sqrt{1-P} < \infty$.

We generalize existing limit theory for stochastic integrals driven by semimartingales and with left-continuous integrands. Joint Skorohod convergence is replaced with joint finite dimensional convergence plus an assumption excluding the case when oscillations of the integrand appear immediately before oscillations of the integrator. Integrands may converge in a very weak topology. It is also proved that convergence of integrators implies convergence of stochastic integrals with respect to the same topology.

We consider random walks with Gaussian distribution of summands. New representations for Wiener-Hopf factorization components are obtained. The factorization method is used to study the distribution of the excess over one-sided and two-sided boundaries. Asymptotic expansions for these distributions and for the expectation of the first exit time are obtained under the assumption that the boundaries tend to infinity.

The Wills functional from the theory of lattice point enumeration can be adapted to produce the following exponential inequality for zero-mean Gaussian processes: $$E \exp [\sup_t (X_t - (1/2) \sigma_t^2)] \leq \exp (E \sup_t X_t).$$

An application is a new proof of the deviation inequality for the supremum of a Gaussian process above its mean:

$$P(\sup_t X_t - E \sup_t X_t \geq a) \leq \exp (-\frac{(1/2) \alpha^2}{\sigma^2}),$$

where $a > 0$ and $\sigma^2 = \sup_t \sigma_t^2$.