We prove a Chung's law of the iterated logarithm for recurrent linear Markov processes. In order to attain this level of generality, our normalization is random. In particular, when the Markov process in question is a diffusion, we obtain the integral test corresponding to a law of the iterated logarithm due to Knight.

Let $B$ be a Brownian motion in $R^d$, $d=2,3$. A time $t\in [0,1]$ is called a cut time for $B[0,1]$ if $B[0,t) \cap B(t,1] = \emptyset$. We show that the Hausdorff dimension of the set of cut times equals $1 - \zeta$, where $\zeta = \zeta_d$ is the intersection exponent. The theorem, combined with known estimates on $\zeta_3$, shows that the percolation dimension of Brownian motion (the minimal Hausdorff dimension of a subpath of a Brownian path) is strictly greater than one in $R^3$.

Let $B$ be a Borel subset of $R^d$ with finite volume. We give an eigenvalue expansion for the transition densities of Brownian motion killed on exiting $B$. Let $A_1$ be the time spent by Brownian motion in a closed cone with vertex $0$ until time one. We show that $\lim_{u\to 0} \log P^0(A_1 < u) /\log u = 1/\xi$ where $\xi$ is defined in terms of the first eigenvalue of the Laplacian in a compact domain. Eigenvalues of the Laplacian in open and closed sets are compared.

A model is proposed for a decreasing sequence of random variables $(V_1, V_2, \cdots)$ with $\sum_n V_n = 1$, which generalizes the Poisson-Dirichlet distribution and the distribution of ranked lengths of excursions of a Brownian motion or recurrent Bessel process. Let $V_n$ be the length of the $n$th longest component interval of $[0,1]\backslash Z$, where $Z$ is an a.s. non-empty random closed of $(0,\infty)$ of Lebesgue measure $0$, and $Z$ is self-similar, i.e. $cZ$ has the same distribution as $Z$ for every $c > 0$. Then for $0 \le a < b \le 1$ the expected number of $n$'s such that $V_n \in (a,b)$ equals $\int_a^b v^{-1} F(dv)$ where the structural distribution $F$ is identical to the distribution of $1 - \sup ( Z \cap [0,1] )$. Then $F(dv) = f(v)dv$ where $(1-v) f(v)$ is a decreasing function of $v$, and every such probability distribution $F$ on $[0,1]$ can arise from this construction.

We introduce an interacting random process related to Ulam's problem, or finding the limit of the normalized longest increasing subsequence of a random permutation. The process describes the evolution of a configuration of sticks on the sites of the one-dimensional integer lattice. Our main result is a hydrodynamic scaling limit: The empirical stick profile converges to a weak solution of the inviscid Burgers equation under a scaling of lattice space and time. The stick process is also an alternative view of Hammersley's particle system that Aldous and Diaconis used to give a new solution to Ulam's problem. Along the way to the scaling limit we produce another independent solution to this question. The heart of the proof is that individual paths of the stochastic process evolve under a semigroup action which under the scaling turns into the corresponding action for the Burgers equation, known as the Lax formula. In a separate appendix we use the Lax formula to give an existence and uniqueness proof for scalar conservation laws with initial data given by a Radon measure.

A countable system of linearly interacting diffusions on the interval [0,1], indexed by a hierarchical group is investigated. A particular choice of the interactions guarantees that we are in the diffusive clustering regime, that is spatial clusters of components with values all close to 0 or all close to 1 grow in various different scales. We studied this phenomenon in [FG94]. In the present paper we analyze the evolution of single components and of clusters over time. First we focus on the time picture of a single component and find that components close to 0 or close to 1 at a late time have had this property for a large time of random order of magnitude, which nevertheless is small compared with the age of the system. The asymptotic distribution of the suitably scaled duration a component was close to a boundary point is calculated. Second we study the history of spatial 0- or 1-clusters by means of time scaled block averages and time-space-thinning procedures. The scaled age of a cluster is again of a random order of magnitude. Third, we construct a transformed Fisher-Wright tree, which (in the long-time limit) describes the structure of the space-time process associated with our system. All described phenomena are independent of the diffusion coefficient and occur for a large class of initial configurations (universality).

The question considered in this paper is which sequences of $p$-integrable random variables can be represented as conditional expectations of a fixed random variable with respect to a given sequence of sigma-fields. For finite families of sigma-fields, explicit inequality equivalent to solvability is stated; sufficient conditions are given for finite and infinite families of sigma-fields, and explicit expansions are presented.

In this paper we shall discuss the almost sure exponential stability for a neutral differential difference equation with damped stochastic perturbations of the form $d[x(t)-G(x(t-\tau))] = f(t,x(t),x(t-\tau))dt + \sigma(t) dw(t)$. Several interesting examples are also given for illustration. It should be pointed out that our results are even new in the case when $\sigma(t) \equiv 0$, i.e. for deterministic neutral differential difference equations.

We develop quantitative bounds on rates of convergence for continuous-time Markov processes on general state spaces. Our methods involve coupling and shift-coupling, and make use of minorization and drift conditions. In particular, we use auxiliary coupling to establish the existence of small (or pseudo-small) sets. We apply our method to some diffusion examples. We are motivated by interest in the use of Langevin diffusions for Monte Carlo simulation.

We study the metastability of the stochastic three dimensional Ising model on a finite torus under a small positive magnetic field at very low temperatures.

We consider the Skorokhod equation $$dX_t=dW_t+(1/2)\nu(X_t), dL_t$$ in a domain $D$, where $W_t$ is Brownian motion in $R^d$, $\nu$ is the inward pointing normal vector on the boundary of $D$, and $L_t$ is the local time on the boundary. The solution to this equation is reflecting Brownian motion in $D$. In this paper we show that in Lipschitz domains the solution to the Skorokhod equation is unique in law.

The three-color Greenberg--Hastings model (GHM) is a simple cellular automaton model for an excitable medium. Each site on the lattice $Z^2$ is initially assigned one of the states 0, 1 or 2. At each tick of a discrete--time clock, the configuration changes according to the following synchronous rule: changes $1\to 2$ and $2\to 0$ are automatic, while an $x$ in state 0 may either stay in the same state or change to 1, the latter possibility occurring iff there is at least one representative of state 1 in the local neighborhood of $x$. Starting from a product measure with just 1's and 0's such dynamics quickly die out (turn into 0's), but not before 1's manage to form infinite connected sets. A very precise description of this ``transient percolation'' phenomenon can be obtained when the neighborhood of $x$ consists of 8 nearest points, the case first investigated by S. Fraser and R. Kapral. In addition, first percolation times for related monotone models are addressed.

Let $S(n)$ be a simple random walk taking values in $Z^d$. A time $n$ is called a cut time if \[ S[0,n] \cap S[n+1,\infty) = \emptyset . \] We show that in three dimensions the number of cut times less than $n$ grows like $n^{1 - \zeta}$ where $\zeta = \zeta_d$ is the intersection exponent. As part of the proof we show that in two or three dimensions \[ P(S[0,n] \cap S[n+1,2n] = \emptyset ) \sim n^{-\zeta}, \] where $\sim$ denotes that each side is bounded by a constant times the other side.

We study a class of systems of countably many linearly interacting diffusions whose components take values in $[0, \inf)$ and which in particular includes the case of interacting (via migration) systems of Feller's continuous state branching diffusions. The components are labelled by a hierarchical group. The longterm behaviour of this system is analysed by considering space-time renormalised systems in a combination of slow and fast time scales and in the limit as an interaction parameter goes to infinity. This leads to a new perspective on the large scale behaviour (in space and time) of critical branching systems in both the persistent and non-persistent cases and including that of the associated historical process. Furthermore we obtain an example for a rigorous renormalization analysis.