The random-cluster model is a generalization of percolation and ferromagnetic Potts models, due to Fortuin and Kasteleyn. Not only is the random-cluster model a worthwhile topic for study in its own right, but also it provides much information about phase transitions in the associated physical models. This paper serves two functions. First, we introduce and survey random-cluster measures from the probabilist's point of view, giving clear statements of some of the many open problems. Second, we present new results for such measures, as follows. We discuss the relationship between weak limits of random-cluster measures and measures satisfying a suitable DLR condition. Using an argument based on the convexity of pressure, we prove the uniqueness of random-cluster measures for all but (at most) countably many values of the parameter $p$. Related results concerning phase transition in two or more dimensions are included, together with various stimulating conjectures. The uniqueness of the infinite cluster is employed in an intrinsic way in part of these arguments. In the second part of this paper is constructed a Markov process whose level sets are reversible Markov processes with random-cluster measures as unique equilibrium measures. This construction enables a coupling of random-cluster measures for all values of $p$. Furthermore, it leads to a proof of the semicontinuity of the percolation probability and provides a heuristic probabilistic justification for the widely held belief that there is a first-order phase transition if and only if the cluster-weighting factor $q$ is sufficiently large.

This paper deals with first passage percolation where the usual i.i.d. condition is weakened to stationarity (and ergodicity). The well known asymptotic shape result is known to extend to this case. It is easy to give necessary conditions for a compact set $B \subset \mathbb{R}^d$ to arise as the asymptotic shape for some stationary measure on the passage times. Our main result says that these conditions are also sufficient.

The "true" self-avoiding walk with bond repulsion is a nearest neighbor random walk on $\mathbb{Z}$, for which the probability of jumping along a bond of the lattice is proportional to $\exp(-g \cdot$ number of previous jumps along that bond). First we prove a limit theorem for the distribution of the local time process of this walk. Using this result, later we prove a local limit theorem, as $A \rightarrow \infty$, for the distribution of $A^{-2/3}X_{\theta_{s/A}}$, where $\theta_{s/A}$ is a random time distributed geometrically with mean $e^{-s/A}(1 - e^{-s/A})^{-1} = A/s + O(1)$. As a by-product we also obtain an apparently new identity related to Brownian excursions and Bessel bridges.

A diffusion process with smooth and nondegenerate elliptic infinitesimal generator on a manifold $M$ induces a Riemannian metric $g$ on $M$. This paper discusses in detail different symmetry properties of such a diffusion by geometric methods. Partial differential equations associated with the generator are studied likewise. With an eye to modelling and applications to filtering, relationships between symmetries of deterministic systems and symmetries of diffusion processes are delineated. The incidence of a stochastic framework on the properties of an original deterministic system are then illustrated in different examples. The construction of a diffusion process with given symmetries is also addressed and resulting geometric problems are raised.

We study certain classes of exceptional times of a linear Brownian motion $(B_t, t \geq 0)$. In particular, we consider the set $K^-$ of all instants $t \in \lbrack 0, 1\rbrack$ such that the value $B_t$ of the Brownian motion at time $t$ is greater than its mean value over all intervals $\lbrack s,t\rbrack, s < t$. We also study the subset $K$ of $K^-$ of all instants $t$ such that in addition $B_t$ is greater than the mean value of $B$ over the intervals $\lbrack t,s\rbrack, t < s \leq 1$. We compute the Hausdorff dimension of $K^-, K$ and some other related sets of exceptional times. These results are closely related to a recent work of Sinai motivated by the analysis of solutions to the Burgers equation with random initial data. The proofs involve studying suitable approximations of the sets $K^-$ and $K$, and deriving precise estimates for the probability that a given time $t$ belongs to these approximations. A delicate zero-one law argument is also needed to prove that the lower bound on the dimension of $K$ holds with probability 1.

Suppose $\varepsilon \in \lbrack 0,1)$ and let $\theta_\varepsilon(t) = (1 - \varepsilon)\sqrt{2t\ln_2 t}$. Let $L^\varepsilon_t$ denote the amount of local time spent by Brownian motion on the curve $\theta_\varepsilon(s)$ before time $t$. If $\varepsilon > 0$, then $\lim\sup_{t\rightarrow\infty} L^\varepsilon_t/\sqrt{2t\ln_2 t} = 2\varepsilon + o(\varepsilon)$. For $\varepsilon = 0$, a nontrivial lim sup is obtained when the normalizing function $\sqrt{2t\ln_2t}$ is replaced by $g(t) = \sqrt{t/\ln_2 t}\ln_3 t$.

We study the asymptotics of the exit probability $\mathbb{P}^\varepsilon_{x,s}\{\tau \leq T\}$, where $\tau$ is the exit time from an open set and $\mathbb{P}^\varepsilon_{x,s}$ is the law of a diffusion process with a small parameter $\varepsilon$ multiplying the diffusion coefficient. We consider the case of the Brownian bridge in many dimensions, this choice being motivated by applications to numerical simulation. The method uses recent results reducing the problem to the solution of a system of linear first-order PDE's.

General characterizations of geometric convergence for Markov chains in discrete time on a general state space have been developed recently in considerable detail. Here we develop a similar theory for $\varphi$-irreducible continuous time processes and consider the following types of criteria for geometric convergence: 1. the existence of exponentially bounded hitting times on one and then all suitably "small" sets; 2. the existence of "Foster-Lyapunov" or "drift" conditions for any one and then all skeleton and resolvent chains; 3. the existence of drift conditions on the extended generator $\tilde\mathscr{A}$ of the process. We use the identity $\tilde\mathscr{A}R_\beta = \beta(R_\beta - I)$ connecting the extended generator and the resolvent kernels $R_\beta$ to show that, under a suitable aperiodicity assumption, exponential convergence is completely equivalent to any of criteria 1-3. These conditions yield criteria for exponential convergence of unbounded as well as bounded functions of the chain. They enable us to identify the dependence of the convergence on the initial state of the chain and also to illustrate that in general some smoothing is required to ensure convergence of unbounded functions.

We consider the one-dimensional catalytic branching process introduced by Dawson and Fleischmann, which is a modification of the super-Brownian motion. The catalysts are given by a nonnegative infinitely divisible random measure with independent increments. We give sufficient conditions for the global support of the process to be compact, and sufficient conditions for noncompact global support. Since the catalytic process is related to the heat equation, compact support may be surprising. On the other hand, the super-Brownian motion has compact global support. We find that all nonnegative stable random measures lead to compact global support, and we give an example of a very rarified Levy process which leads to noncompact global support.

We show that two-dimensional super-Brownian motion is a multiple of the $h$-Hausdorff measure on its closed support, where $h(r) = r^2 \log^+(1/r)\log^+ \log^+ \log^+(1/r)$. This complements known results in dimensions greater than 2.

Consider the supercritical super-Brownian motion $X(t,\cdot)$ on $R^d$ corresponding to the evolution equation $u_t = \frac{D}{2}\Delta u + u - u^2.$ We obtain rather tight bounds on $P_\mu(X(s,B^c_n(0)) = 0$, for all $s \in \lbrack 0,t\rbrack)$ and on $P_\mu(X(t,B^c_n(0)) = 0)$, for large $n$, where $P_\mu$ denotes the measure corresponding to the supercritical super-Brownian motion starting from the finite measure, $\mu, B_n(0) \subset R^d$ denotes the ball of radius $n$ centered at the origin and $B^c_n(0)$ denotes its complement. In particular, we show, for example, that if $\mu$ is a compactly supported, finite measure on $R^d$, then $\lim_{n\rightarrow\infty} P_\mu(X(t,B^c_n(0)) = 0, \text{for all} t \in \lbrack 0,\gamma n\rbrack) = 1 \text{if} \gamma < (2D)^{-1/2}$ and $\lim_{n\rightarrow\infty} P_\mu(X(\gamma n, B^c_n(0)) = 0\mid\text{the process survives}) = 0 \text{if} \gamma > (2D)^{-1/2}.$

Let $\mu_t(dx)$ denote a three-dimensional super-Brownian motion with deterministic initial state $\mu_0(dx) = dx$, the Lebesgue measure. Let $V: \mathbb{R}^3 \mapsto \mathbb{R}$ be Holder-continuous with compact support, not identically zero and such that $\int_{\mathbb{R}^3}V(x)dx = 0$. We show that $\log P\big\{\int^t_0\int_{\mathbb{R}^3} V(x)\mu_s(dx) ds > bt^{3/4}\big\}$ is of order $t^{1/2}$ as $t \rightarrow \infty$, for $b > 0$. This should be compared with the known result for the case $\int_{\mathbb{R}^3}V(x)dx > 0$. In that case the normalization $bt^{3/4}, b > 0$, must be replaced by $bt, b > \int_{\mathbb{R}^3}V(x)dx$, in order that the same statement hold true. While this result only captures the logarithmic order, the method of proof enables us to obtain complete results for the corresponding moderate deviations and central limit theorems.

It is known from previous work of the authors that any square-integrable functional of a superprocess may be represented as a constant plus a stochastic integral against the associated orthogonal martingale measure. Here we give, for a large class of such functionals, an explicit description of the integrand that is analogous to Clark's formula for the representation of certain Brownian functionals. As a consequence, we develop a partial analogue of the Wiener chaos expansion in the superprocess setting. Rather than work with superprocesses per se, our results are stated and proved in the richer and more natural context of the associated historical process.

Duality has proved to be a powerful tool in the theory of interacting particle systems. The approach in this paper is algebraic rather than via Harris diagrams. A form of duality is found which includes coalescing and annihilating duality as special cases. This enables new results for the branching annihilating random walk and the biased annihilating branching process to be derived.

We discuss the hydrodynamic scaling limits starting with deterministic configurations for different models. Certain estimates on the entropy of the system are derived.

In this paper we prove some intermittency-type estimates for the stochastic partial differential equation $du = \mathscr{L}u dt + \mathscr{M}_lu\circ dW^l_t$, where $\mathscr{L}$ is a strongly elliptic second-order partial differential operator and the $\mathscr{M}_l$'s are first-order partial differential operators. Here the $W^l$'s are standard Wiener processes and $\circ$ denotes Stratonovich integration. We assume for simplicity that $u(0,\cdot) \equiv 1$. Our interest here is the behavior of $\mathbb{E}\lbrack|u(t,x)|^p\rbrack$ for large time and large $p$; more specifically, our interest is the growth of $(p^2t)^{-1}\log\mathbb{E}\lbrack|u(t,x)|^p\rbrack$ as $t$, then $p$, become large.

We establish the existence of smooth densities for solutions of $\mathbf{R}^d$-valued stochastic hereditary differential systems of the form $dx(t) = H(t,x)dt + g(t,x(t - r))dW(t).$ In the above equation, $W$ is an $n$-dimensional Wiener process, $r$ is a positive time delay, $H$ is a nonanticipating functional defined on the space of paths in $\mathbf{R}^d$ and $g$ is an $n \times d$ matrix-valued function defined on $\lbrack 0,\infty) \times \mathbf{R}^d$, such that $gg^\ast$ has degeneracies of polynomial order on a hypersurface in $\mathbf{R}^d$. In the course of proving this result, we establish a very general criterion for the hypoellipticity of a class of degenerate parabolic second-order time-dependent differential operators with space-independent principal part.

We consider the Zakai equation for the unnormalized conditional distribution $\sigma$ when the signal process $X$ takes values in a complete separable metric space $E$ and when $h$ is a continuous, possibly unbounded function on $E$. It is assumed that $X$ is a Markov process which is characterized via a martingale problem for an operator $A_0$. Uniqueness of solution for the measure-valued Zakai and Fujisaki-Kallianpur-Kunita equations is proved when the test functions belong to the domain of $A_0$. It is also shown that the conditional distributions are robust.

This paper gives sharp bounds on the eigenvalues of a natural random walk on the group of upper triangular $n \times n$ matrices over the field of characteristic $p$, an odd prime, with 1's on the diagonal. In particular, this includes the finite Heisenberg groups as a special case. As a consequence we get bounds on the time required to achieve randomness for these walks. Some of the steps are done using the geometric bounds on the eigenvalues of Diaconis and Stroock. However, the crucial step is done using more subtle and idiosyncratic techniques. We bound the eigenvalues inductively over a sequence of subspaces.

In this paper we give sharp bounds on the eigenvalues of the natural random walk on the Burnside group $B(3,n)$. Most of the argument uses established geometric techniques for eigenvalue bounds. However, the most interesting bound, the upper bound on the second largest eigenvalue, cannot be done by existing techniques. To give a bound we use a novel method for bounding the eigenvalues of a random walk on a group $G$ (or equivalently its Cayley graph). This method works by choosing eigenvectors which fall into representations of an Abelian normal subgroup of $G$. One is then left with a large number (one for each representation) of easier problems to analyze.

In this paper we discuss and apply a novel method for bounding the eigenvalues of a random walk on a group $G$ (or equivalently on its Cayley graph). This method works by looking at the action of an Abelian normal subgroup $H$ of $G$ on $G$. We may then choose eigenvectors which fall into representations of $H$. One is then left with a large number (one for each representation of $H$) of easier problems to analyze. This analysis is carried out by new geometric methods. This method allows us to give bounds on the second largest eigenvalue of random walks on nilpotent groups with low class number. The method also lets us treat certain very easy solvable groups and to give better bounds for certain nice nilpotent groups with large class number. For example, we will give sharp bounds for two natural random walks on groups of upper triangular matrices.

This paper is motivated by remarkable results of Mandelbaum, Shepp and Vanderbei concerning an optimal switching problem for two Brownian motions. In this paper, the discrete form of this problem, in which the Brownian motions are replaced by random walks, is studied and solved without any restriction on the boundary data. The method proposed here involves uncovering the structure of the solution using combinatorial and geometric arguments, and then providing a characterization for the two types of possible solutions, as well as explicit formulas for computing the solution. The extension of these methods and results to the continuous time problem will be considered in a subsequent paper.

Let $D$ be a compact, convex domain in $d$-dimensional Euclidean space and let $f$ be a nonnegative real-valued function defined on $D$. The classical optimal stopping problem is to find a stopping time $\tau^\ast$ that attains the supremum $\nu(x) = \sup_\tau EE_xf(B(\tau))$. Here, $B$ is a $d$-dimensional Brownian motion with absorption on the boundary of $D$ and the supremum is over all stopping times. It is well known that $\nu$ is characterized as the smallest superharmonic majorant of $f$. In this paper, we modify this problem by allowing $B$ to be essentially any drift-free diffusion (with absorption, as before, on the boundary of $D$). For example, it could be a Brownian motion diffusing on some lower dimensional affine set. In addition, one is allowed to switch among these diffusions at any time. The problem is to find a stopping time and a switching strategy that together attain the supremum over all stopping times and all switching strategies. For this problem, we show that $\nu$ is characterized as the smallest concave majorant of $f$. The domain $D$ can be decomposed into a disjoint union of relatively open convex sets on each of which the function $\nu$ is affine. Furthermore, the union of the zero-dimensional convex sets is contained in the set on which $\nu = f$. An optimal switching strategy is any strategy that at all times diffuses in the affine hull of the current convex set. When the diffusion reaches the boundary of the current convex set, it will lie on a lower dimensional convex set and must then diffuse on the affine hull of this new set. This process continues until the set on which $\nu = f$ is reached, which is the optimal stopping time.

Let $T_r$ be the first time a sum $S_n$ of nondegenerate i.i.d. random variables leaves a ball of radius $r$ in some given norm on $\mathbb{R}^d$. In the case of the Euclidean norm we completely characterize $L^p$-boundedness of the overshoot $\|S_{T_r}\| - r$ in terms of the underlying distribution. For more general norms we provide a similar characterization under a smoothness condition on the norm which is shown to be very nearly sharp. One of the key steps in doing this is a characterization of the possible limit laws of $S_{T_r}/\|S_{T_r}\|$ under the weaker condition $\|S_{T_r}\|/r \rightarrow_p 1$.