The voter model is one of the standard interacting particle systems. Two related problems for this process are to analyze its behavior, after large times $t$, for the sets of sites (1) sharing the same opinion as the site 0, and (2) having the opinion that was originally at 0. Results on the sizes of these sets were given by Sawyer (1979)and Bramson and Griffeath (1980). Here, we investigate the spatial structure of these sets in $d \geq 2$, which we show converge to quantities associated with super-Brownian motion, after suitable normalization. The main theorem from Cox, Durrett and Perkins (2000) serves as an important tool for these results.

Dans cet article, nous donnons des conditions pour qu’il existe une chaîne de Markov indexée par $\mathbf{-N}$ dont le noyau de transition est donné. Lorsque le noyau est irréductible et lorsque de telles chaînes existent, nous décrivons leurs comportements extrémaux. Nous montrons qu’il n’y a que deux types de comportements possibles: un comportement de type stationnaire et un comportement de type transient, où le temps est une fonction déterministe de la position jusqu’àun instant aléatoire strictement supérieur à$-\infty$. Nous donnons des exemples illustrant ces situations.

Limit laws for three-dimensional super-Brownian motion are derived, conditioned on survival up to a large time. A large deviation principle is proved for the joint behavior of occupation times and their difference. These are done via analyzing the generating function and exploiting a connection between probability and differential–integral equations.

A Poisson equation in $\mathbb{R}^d$ for the elliptic operator corresponding to an ergodic diffusion process is considered. Existence and uniqueness of its solution in Sobolev classes of functions is established along with the bounds for its growth. This result is used to study a diffusion approximation for two-scaled diffusion processes usingthe method of corrector; the solution of a Poisson equation serves as a corrector.

The paper studies the behavior of the specific entropy for one-dimensional simple exclusion processes under the hydrodynamic scaling of time and space. It is shown that if the initial configurations possess a macroscopic profile then for each positive macroscopic time the specific microscopic entropy converges to the macroscopic entropy .The latter is defined in terms of the solution of the corresponding hydrodynamic equation.

Consider a process $X(\cdot) = {X(t), 0 \leq t < \infty}$ with values in the interval $I = (0, 1)$, absorption at the boundary points of $I$, and dynamics

$$dX(t) = \beta(t)dt + \sigma(t)dW(t),\quad X(0) = x.$$

The values $(\beta(t), \sigma(t))$ are selected by a controller from a subset of $\Re \times (0, \infty)$ that depends on the current position $X(t)$, for every $t \geq 0$. At any stopping rule $\tau$ of his choice, a second player, called a stopper, can halt the evolution of the process $X(\cdot)$, upon which he receives from the controller the amount $e^{-\alpha\tau}u(X(\tau))$; here $\alpha \epsilon [0, \infty)$ is a discount factor, and $u: [0, 1] \to \Re$ is a continuous “reward function.” Under appropriate conditions on this function and on the controller’s set of choices, it is shown that the two players have a saddlepoint of “optimal strategies.” These can be described fairly explicitly by reduction to a suitable problem of optimal stopping, whose maximal expected reward V coincides with the value of the game,

$$V = \sup_{\tau} \inf_{X(\cdot)} \mathbf{E}[e^{-\alpha\tau}u(X(\tau))] = \inf_{X(\cdot)} \sup_{\tau} \mathbf{E}[e^{-\alpha\tau}u(X(\tau))].$$

We define a scaling limit of the height function on the domino tiling model (dimer model) on simply connected regions in $\mathbf{Z}^2$ and show that it is the “massless free field,” a Gaussian process with independent coefficients when expanded in the eigenbasis of the Laplacian.

We study the large scale space–time fluctuations of an interface which is modeled by a massless scalar field with reversible Langevin dynamics. For a strictly convex interaction potential we prove that on a large space–time scale these fluctuations are governed by an infinite-dimensional Ornstein –Uhlenbeck process. Its effective diffusion type covariance matrix is characterized through a variational formula.

We derive a large deviation principle both quenched and annealed for a one-dimensional diffusion process in a drifted Brownian environment providing the continuous time analogue of what Comets, Gantert and Zeitouni recently establish for the random walk in random environment. A key-ingredient, Kotani’s lemma, allows us to compute the corresponding rate functions. The results are more explicit than in the discrete-time setting.

Using Itô’s calculus, we study the large deviations properties of the law of the spectral measure of the Hermitian Brownian motion.We extend this strategy to the symmetric, unitary and Wishart processes. This dynamical approach is generalized to the study of the large deviations of the non-commutative laws of several independent Hermitian Brownian motions. As a consequence, we can bound from above entropies defined in the spirit of the microstates entropy introduced by Voiculescu.

We consider the $d$-dimensional voter model for $d \geq 3$. Our interest is the large scale limit of the equilibrium state of the voter model, where we prove the $d = 3$ results of [1] for $d \geq 4$, which turn out to be of a different nature than for $d = 3$. For this purpose we use the historical process.We establish some surprising facts about the Green’s function of random walks in dimension $d \geq 4$,which lead to the different features in $d = 3$ versus $d \geq 4$. Secondly, we prove an analogous result for the voter model on the hierarchical group.

Given two Polish spaces $A_X$ and $A_Y$, let $\rho : A_X \times A_Y \to \mathbb{R}^d$ be a bounded measurable function. Let $X = {X_n : n \geq 1}$ and $Y = {Y_n : n \geq 1}$ be two independent stationary processes on $A_X^{\infty}$ and $A_Y^{\infty}$, respectively. The article studies the large deviation principle (LDP) for $n^{-1} \sum_{k=1}^n \rho(X_k, Y_k)$, conditional on $X$. Based on a stochastic version of approximate subadditivity, it is shown that if Y satisfies certain mixing condition, then for almost all random realization $x$ of $X$, the laws of $n^{-1} \sum_{k=1}^n \rho(x_k, Y_k)$ satisfy the conditional LDP with a non-random convex rate funcion. Conditions for the rate function to be non-trivial (that is, not $0/\infty$ function) are also given.

We continue our investigations on a class of growth models introduced in a previous paper. Given a nonnegative function $v: \mathbb{Z}^d \to \mathbb{Z}$ with $v(0) = 0$, we define the space of configurations $\Gamma$ to consist of functions $h: \mathbb{Z}^d \to \mathbb{Z}$ such that $h(i) - h(j) \leq v(i - j)$ for all i$i, j \epsilon \mathbb{Z}^d$. We then take two sequances of independent Poisson clocks $(p^{\pm}(i, t): i \epsilon \mathbb{Z}^d)$ of rates $\lambda^{\pm}$. We start with a possibly random configuration $h \epsilon \Gamma$. The function $h$ increases (respectively, decreases) by one unit at site $i$, when the clock $p^{+}(i, \cdot)$ [respectively, $p^{-}(i, \cdot)$] rings and the resulting configuration is still in $\Gamma$. Otherwise the change in h is suppressed. In this way we have a process $h(i, t)$ that after a rescaling $u^{\varepsilon}(x, t) = \varepsilonh([\frac{x}{\varepsilon}], \frac{t}{\varepsilon})$ is expected to converge to a function $u(x, t)$ that solves a Hamilton-Jacobi equation of the form $u_t + H(u_x) = 0$. We established this when $\lambda^{-}$ or $\lambda^{+} = 0$ in the previous paper, employing a strong monotonicity property of the process $h(i, t)$. Such property is no longer available when both $\lambda^{+}, \lambda^{-}$ are nonzero. In this paper we initiate a new approach to treat the problem when th dimension is 1 and the set $\Gamma$ can be described by local constraints on the configuration $h$. In higher dimensions, we can only show that any limit point of th processes $u^{varepsilon}$ is a process $u$ that satisfies a Hamilton-Jacobi equation for a suitable (possibly random) Hamiltonian $H$.

Barbour introduced a probabilistic view of Stein's method for estimating the error in probability approximations. However, in the case of approximations by general distributions on the integers, there have been no purely probabilistic proofs of Stein's bounds till this paper. Furthermore, the methods introduced here apply to a very large class of approximating distributions on the non-negative integers, among which there is a natural class for higher-order approximations by probability distributions rather than signed measures (as previously). The methods also produce Stein magic factors for process approximations which do not increase with the window of observation and which are simpler to apply than those in Brown, Weinberg and Xia.