We consider isotropic stochastic flows in a Euclidean space of $d$ dimensions, $d \geq 2$. The tendency of two-point distances and of tangent vectors to shrink or expand is related to the dimension and the proportion of the flow that is solenoidal or potential. Tangent vectors from the same point tend to become aligned in the same or opposite directions. The purely potential flows are characterized by an analogue of the curl-free property. Liapounov exponents are treated briefly. The rate of increase or decrease of the length of an arc of small diameter is related to the shape of the arc. In the case $d = 2$ a sufficient condition is given under which the length of a short arc has a high probability of approaching 0.

Consider the Feynman-Kac functional $u(q, D; x) = E_x\exp(\int^{\tau_D}_0 q(x(s)) ds),$ where $D$ is a bounded open region in $R^d, \tau_D$ is the first exit time from $D, q \in C(\bar{D})$, and $x(s)$ is a diffusion process on $R^d$ with generator $L$. We give a criterion for the finiteness or infiniteness of $u(q, D; x)$ in terms of the top of the spectrum of the Schrodinger operator $L_{q, D}$, an extension of $L + q$ acting on smooth functions which vanish on $\partial D$. As we also have a variational formula for the top of the spectrum, we thus obtain a criterion explicitly in terms of a variational formula.

It is shown that if a diffusion process, $\{X_t: 0 \leq t \leq 1\}$, on $R^d$ satisfies $dX_t = b(t, X_t) dt + \sigma (t, X_t) dw_t$ then the reversed process, $\{\bar{X}_t: 0 \leq t \leq 1\}$ where $\bar{X}_t = X_{1 - t}$, is again a diffusion with drift $\bar{b}$ and diffusion coefficient $\bar{\sigma}$, provided some mild conditions on $b, \sigma$, and $p_0$, the density of the law of $X_0$, hold. Moreover $\bar{b}$ and $\bar\sigma$ are identified.

We consider a one-dimensional diffusion process in a random medium, which is itself generated by a Wiener process. The asymptotic behaviour is investigated, using the self-similarity of the medium process. The results are analogous to the lattice case.

Let $B$ be a Brownian motion with values in Euclidean space $R^d$, where $d = 2 \text{or} 3$. The Wiener sausage with radius $\varepsilon$ associated with $B$ is defined as the set of points whose distance from the path is less than $\varepsilon$. Let $B'$ be another Brownian motion with values in $R^d$, independent of $B$. The Lebesgue measure of the intersection of the Wiener sausages associated with $B$ and $B'$, suitably normalized, is shown to converge, when $\varepsilon$ goes to 0, towards the intersection local time of $B$ and $B'$, as defined by German, Horowitz and Rosen. This approximation of the intersection local time is used to prove a conjecture of Taylor, relating to the Hausdorff measure of the set of multiple points of planar Brownian motion.

We use a Tanaka-like formula to explain Varadhan's renormalization of the formally infinite measure of Brownian self intersections given by $\int^T_0 \int^T_0 \delta(W_t - W_s) ds dt.$

Let $W(t), 0 \leq t < \infty$, be a Wiener process. This paper proves that $\lim \sup_{T \rightarrow \infty} \sup_{0 < t \leq T} \frac{|W(T) - W(T - t)|}{\{2t(\log(T/t) + \log\log t)\}^{1/2}} = 1, a.s.,$ $\lim_{T \rightarrow \infty} \sup_{0 < t \leq T} \sup_{t \leq s \leq T} \frac{|W(T) - W(s - t)|}{\{2t(\log(T/t) + \log \log t)\}^{1/2}} = 1, a.s.$ These results give an affirmative answer to the questions posed by Hanson and Russo without additional assumptions.

It is shown that one-dimensional Gibbs states may be represented as concatenations of infinite lists of iid "words." It follows that Gibbs states inherit many properties of recurrent Markov chains on denumerable state spaces.

Let $\{N(t): t \geq 0\}$ be a renewal counting process with lifetime density $f(t)$. For each bounded Borel set $A$ contained in $\lbrack 0, \infty)$, denote the number of renewals in $A$ by $N(A)$. The renewal process is called associated if the corresponding family of random variables, $N(A)$, is associated. The result of this note is that the renewal process is associated whenever $\log(f)$ is a convex function (which implies a decreasing failure rate).

The simple exclusion process "as seen from a tagged particle" is studied. The set of translation invariant and invariant measures for this process is determined in the translation invariant case on $\mathbb{Z}^d$. The set of all invariant measures is determined in the nearest neighbor asymmetric case on $\mathbb{Z}$. The domains of attraction of the invariant measures are established in the one-dimensional nearest neighbor translation invariant case.

If $(\xi^A(t), t \geq 0)$ is the contact process with initial configuration $A, f: \mathscr{P}(\mathbb{Z}) \rightarrow \mathbb{R}$ is any cylindrical function and $|A| = \infty$, we prove a central limit theorem for $(f(\xi^A(t)), t \geq 0)$ when the rate of infection is supercritical.

Let $\{S_n, n \geq 1\}$ be a random walk and $N$ a stopping time. The Burkholder-Gundy-Davis inequalities for martingales can be used to give conditions on the moments of $N$ (and of $X = S_1$), which ensure the finiteness of the moments of the stopped random walk, $S_N$. We establish converses to these results, that is, we obtain conditions on the moments of the stopped random walk and $X$ or $N$ which imply the finiteness of the moments of $N$ or $X$. We also study one-sided versions of these problems and corresponding questions concerning uniform integrability (of families of stopping times and families of stopped random walks).

Let $\{m_{ij}\}, i = 1,2,\ldots, j = 1,2,\ldots,$ be iid random variables with $Em_{11} = 0$ and $Em^2_{11} = \sigma^2$. For each $n$ define $M_n = \{m_{ij}\}_{1 \leq i, j \leq n}$, the $n \times n$ matrix whose $(i, j)$ component is $m_{ij}$. We show that $\lim \sup_{n \rightarrow \infty}\rho_n \leq \sigma$ a.s., where $\rho_n$ is the spectral radius of $M_n/\sqrt n$. Evidence from computer experiments indicates that in fact $\rho_n \rightarrow \sigma$ a.s.

Necessary and sufficient conditions on $P$ for the unit balls of $BL(\mathbf{R})$ and $\operatorname{Lip}(\mathbf{R})$ to be functional $P$-Donsker classes are obtained.

In this paper we compute exact rates at which the sample extremes become independent uniformly over all Borel sets.

The asymptotic distribution of a "triangular" scheme of $U$-statistics is studied. Two limit theorems, applicable in different situations, are given. One theorem yields convergence to a normal distribution; the other includes Poisson limits and other limit laws. Applications to statistics based on small interpoint distances in a sample are given.

Let $\{X_j, j \geq 1\}$ be a strictly stationary sequence of random variables with mean zero, finite variance, and satisfying a strong mixing condition. Denote by $S_n$ the $n$th partial sum and suppose that $\operatorname{Var} S_n$ is regularly varying of order 1. We prove that if $S_n (\operatorname{Var} S_n)^{-1/2}$ does not converge to zero in $L^1$, then $\{X_j, j \geq 1\}$ is in the domain of partial attraction of a Gaussian law. If, however, no subsequence of $\{S_n(\operatorname{Var} S_n)^{-1/2}, n \geq 1\}$ converges to zero in $L^1$ and if $E|S_n|$ is regularly varying of order $\frac{1}{2}$, then $\{X_j, j \geq 1\}$ is in the domain of attraction to a Gaussian law. In each case the norming constant can be chosen as $E|S_n|$.

Let $\{Z_i: -\infty < i < + \infty\}$ be a strictly stationary $\alpha$-mixing sequence. Without specifying the dependence model giving rise to $\{Z_i\}$, and without specifying the marginal distribution of $Z_i$, we address the question of asymptotic normality for a general statistic $t_n(Z_1,\ldots, Z_n)$. The main theoretical result is a set of necessary and sufficient conditions for joint asymptotic normality of $t_n$ and a subseries value $t_m (m \leq n)$. Our theorems on asymptotic normality are the natural analogs to earlier results that deal with general statistics from iid sequences, and to other results that apply to the sample mean from dependent sequences. Asymptotic normality of the sample mean and of the sample fractiles follows as a special case of our general statistic $t_n$.

Watanabe proved that if $X_t$ is a point process such that $X_t - t$ is a martingale, then $X_t$ is a Poisson process and this result was generalized by Bremaud for doubly stochastic Poisson processes. Here we define two-parameter point processes and extend this property without needing the strong martingale condition. Using this characterization, we study the problem of transforming a two-parameter point process into a two-parameter Poisson process by means of a family of stopping lines as a time change. Nualart and Sanz gave conditions in order to transform a square integrable strong martingale into a Wiener process. Here, we do the same for the Poisson process by a similar method but under more general conditions.

We show that a sum of dependent random variables is approximately compound Poisson when the variables are rarely nonzero and, given they are nonzero, their conditional distributions are nearly identical. We give several upper bounds on the total-variation distance between the distribution of such a sum and a compound Poisson distribution. Included is an example for Markovian occurrences of a rare event. Our bounds are consistent with those that are known for Poisson approximations for sums of uniformly small random variables.

Steck provides several approximations for the multivariate Mills' ratio. We first prove a new result for the univariate Mills' ratio and use it to give simple sufficient conditions for Steck's best approximation to be a lower bound.

Let $A = A(r, 1)$ be an annulus $\{z: r < |z| < 1\}$ with the Poincare metric $g$ on $A$. Let $\mathbf{Z} = (Z_t, P_a)$ be a Brownian motion on $A$ corresponding to $g$. If we take a geodesic disc $D$ centered at $c$ in $A$, then the probability $P_a(\exists t, Z_t \in \partial D$ such that $Z_s, 0 < s < t$, winds around the origin in the positive direction) is a function of $r, |c|$, and the radius $\rho$ of $D$. In the present paper we shall calculate the value $S$ of the supremum of these winding probabilities. Then it will turn out that there exists a 1 to 1 correspondence between $S$ and $r$. Noting that $r$ is called the modulus of $A$, we have an explicit formula of moduli of annular regions. Further we shall give an explicit formula of moduli of tori in a similar way.

A completely simple semigroup has the form $S = X \times G \times Y$. This paper considers the relationship between $S$ and $G$. Given a recurrent random walk on $S$ we determine under what conditions $G$ is also recurrent and conversely. In particular we generalize the results of Larisse.

De Finetti's theorem for stationary Markov exchangeability states that a sequence having a stationary and Markov exchangeable distribution is a mixture of Markov chains. A finite version of this theorem is given by considering a finite sequence $X_1,\ldots, X_n$ which is stationary and Markov exchangeable. It is shown that any portion of $k$ consecutive elements, say $X_1,\cdots, X_k$ for $k < n$, is nearly a mixture of Markov chains (the distance measured in the variation norm).