The Loeb measure construction of nonstandard analysis is used to define uniform probability $\mu_L$ on the infinite-dimensional sphere of Poincare, Wiener and Levy, and we construct Wiener measure from it, thus giving rigorous sense to the informal discussion by McKean. From this follows an elementary proof of a weak convergence result. The relation to the infinite product of Gaussian measures is studied. We investigate transformations of the sphere induced by shifts and the associated transformations of $\mu_L$. The Cameron-Martin density is derived as a Jacobian. We also prove a dichotomy theorem for the family of shifted measures.

The problem of computing the lower tail of a Gaussian norm is considered in this paper. Based on large deviations arguments, a bound on these tails is derived which is tighter than those obtained by other methods. Conditional exponential moment bounds are also derived and as an application, $L^2$ type Onsager-Machlup functionals for diffusions are computed.

Given an $\mathbb{R}$-valued, Borel measurable function $F$ on an abstract Wiener space $(E, H, \mu)$, we show that $F$ is uniformly Lipschitz continuous in the directions of $H$ if and only if it has one derivative in the sense of Malliavin and that derivative is an element of $L^\infty(\mu; H)$.

This paper gives a general two-term approximation for the tail probability of the maxima of a class of differentiable Gaussian random fields and illustrates its potential statistical applications.

Suppose that $P$ and $P_n, n \in \mathscr{N}$, are probabilities on a real, separable Hilbert space, $V$. It is known that if $P$ satisfies some regularity conditions and $X$ is such that $P_X = P$, then there exist mappings $H_n: V \rightarrow V$, such that $P_{H_n(X)} = P_n$ and the Wasserstein distance between $P_n$ and $P$ coincides with $(\int\|x - H_n(x)\|^2 dP)^{1/2}, n \in \mathscr{N}$. In this paper we prove that the weak convergence of $\{P_n\}$ to $P$ is enough to ensure that $\{H_n(X)\}$ converges to $X$ in measure, and that, if $V = \mathcal{R}^p$, then the convergence is also a.e. This property seems to be characteristic of finite-dimensional spaces, because we include an example, with $V$ infinite-dimensional and $P$ Gaussian, where a.e. convergence does not hold.

It is shown that in the case of stationary Gaussian processes, the $J$th $(J \geq 1)$ record times $\{T_n, n \geq 1\}$ and the corresponding $J$-upper order statistics $\{X_{T_n - J + 1, T_n}, \ldots, X_{T_n, T_n}\}$ can almost surely be identified via a translation of the time index $n$ to the corresponding elements defined on a sequence of independent and identically distributed random variables. A construction method for approximating sequences of record times and the corresponding upper order statistics introduced by Haiman (1987a, b) for the case $J = 1$ is extended and applied under weaker conditions concerning the covariance function, and also under different sets of new hypotheses.

In an important paper, Strassen discussed the existence of measures on product spaces with given marginals in the context of Polish spaces. We generalize his results to arbitrary Hausdorff spaces and indicate some applications such as measures with given support and stochastic inequalities on partially ordered Hausdorff spaces. In the second part of our paper, we state two results on the general moment problem, thus generalizing earlier theorems due to Kemperman.

We study the relation between stochastic domination of an infinitely divisible random vector $\mathbf{X}$ by another infinitely divisible random vector $\mathbf{Y}$ and their corresponding Levy measures. The results are used to derive a Slepian-type inequality for a general class of symmetric infinitely divisible random vectors.

Using a result by Landers and Rogge, it is shown that the Hausdorff metric of $\sigma$-fields is uniformly equivalent to the metric induced by the Hausdorff distance between sets of measurable functions. An application is given to the continuity of the value of information with respect to the Hausdorff metric of $\sigma$-fields.

We consider the random variables $\xi(\beta) = \sum^\infty_{n = 0}\beta^n\varepsilon_n$ for $\beta < 1$. We prove that if the $\varepsilon_n$ are i.i.d. random variables with mean zero and variance 1, then a law of the iterated logarithm holds in the sense that the cluster set of $\frac{\sqrt{1 - \beta^2}}{2\log\log(1/(1 - \beta^2))}\xi(\beta),$ when $\beta$ converges to one, is the interval $\lbrack-1, 1\rbrack$.

A class of topological semigroups called GZH-semigroups is introduced. Conditions under which they have the property that limits of infinitesimal arrays are infinitely divisible are obtained. The convolution semigroup of all probability measures on a second countable LCA-group or on a real separable Hilbert space as well as the semigroup of all positive definite kernels defined on a countable set with complex values and with norms not greater than 1 are reduced to an extended form of Delphic semigroups.

This paper strengthens and generalizes some theorems proved earlier by Lynch and Sethuraman on large deviations (LD) for random processes with independent increments.

Assume a sequence of probabilities $\{P_n\}$ has a large deviation rate function $I$. It is proved that $I$ takes a form analogous to a convex conjugate. If $I$ is also assumed convex, then $I$ is a convex conjugate of an explicitly defined function $\psi$. The results are applied to the empirical law of a Markov chain yielding universal bounds on $I$. Examples are given of Markov chains in which the empirical law has a large deviation rate strictly between the given bounds.

We consider threshold voter systems in which the threshold $\tau > n/2$, where $n$ is the number of neighbors, and we present results in support of the following picture of what happens starting from product measure with density $1/2$. The system fixates, that is, each site flips only finitely many times. There is a critical value, $\theta_c$, so that if $\tau = \theta n$ with $\theta > \theta_c$ and $n$ is large then most sites never flip, while for $\theta \in (1/2, \theta_c)$ and $n$ large, the limiting state consists mostly of large regions of points of the same type. In $d = 1, \theta_c \approx 0.6469076$ while in $d > 1, \theta_c = 3/4$.

Let $(\mathscr{R}(k), k \geq 1)$ be random trees with $k$ leaves, satisfying a consistency condition: Removing a random leaf from $\mathscr{R}(k)$ gives $\mathscr{R}(k - 1)$. Then under an extra condition, this family determines a random continuum tree $\mathscr{L}$, which it is convenient to represent as a random subset of $l_1$. This leads to an abstract notion of convergence in distribution, as $n \rightarrow \infty$, of (rescaled) random trees $\mathscr{J}_n$ on $n$ vertices to a limit continuum random tree $\mathscr{L}$. The notion is based upon the assumption that, for fixed $k$, the subtrees of $\mathscr{J}_n$ determined by $k$ randomly chosen vertices converge to $\mathscr{R}(k)$. As our main example, under mild conditions on the offspring distribution, the family tree of a Galton-Watson branching process, conditioned on total population size equal to $n$, can be rescaled to converge to a limit continuum random tree which can be constructed from Brownian excursion.

This paper considers an infinite system of particles on the integers $\mathbb{Z}$ that: (1) step to the right with a random delay, and (2) split or die along the way according to a random law depending on their position. The exponential growth rate of the particle density is computed in the long time limit in the form of a variational formula that can be solved explicitly. The result reveals two phase transitions associated with localization vs. delocalization and survival vs. extinction. In addition, the system exhibits an intermittency effect. Greven and den Hollander considered the more difficult situation where the particles may step both to the left and right, but the analysis of the phase diagram was less complete.

Let $r(n)$ be the radius of the largest disc covered by $S(1),\ldots, S(n)$, where $\{S(k); k = 1, 2,\ldots\}$ is the simple symmetric random walk on $Z^2$. The main result tells us that $r(n) \geq n^{1/50}$ a.s. for all but finitely many $n$.

Let $X$ be a semimartingale, and $S$ its Snell envelope. Under the assumption that $X$ and $S$ are continuous semimartingales in $H^1$, this article obtains a new, maximal, characterisation of $S$, and gives an application to the optimal stopping of functions of diffusions. We present a counterexample to the standard assertion that $S$ is just "a martingale on the go-region and $X$ on the stop-region."

We present a Brownian embedding for a broad class of compensated compound Poisson processes. Applications of this method are discussed for a problem of level crossings, as well as Donsker's invariance type of principles. In particular, we give a central limit theorem for local times.

We develop new tools that enable us to extend the majorizing measure lower bound to a large class of infinitely divisible processes. We show (in a rigorous sense) that the complexity of these processes is dominated by the complexity of the positive infinitely divisible processes.

Let $\sigma(t)$ be an ergodic Markov chain on a finite state space $E$ and for each $\sigma \in E$, define on $\mathbb{R}^d$ the second-order elliptic operator $L_\sigma = \frac{1}{2} \sum^d_{i,j = 1} a_{ij}(x; \sigma)\frac{\partial^2}{\partial x_i\partial x_j} + \sum^d_{i = 1} b_i(x;\sigma)\frac{\partial}{\partial x_i}.$ Then for each realization $\sigma(t) = \sigma(t, \omega)$ of the Markov chain, $L_{\sigma(t)}$ may be thought of as a time-inhomogeneous diffusion generator. We call such a process a diffusion in a random temporal environment or simply a random diffusion. We study the transience and recurrence properties and the central limit theorem properties for a class of random diffusions. We also give applications to the stabilization and homogenization of the Cauchy problem for random parabolic operators.

Let $L$ generate a transient diffusion $X(t)$ on $R^d$ and let $D$ be an exterior domain. Let $h$ be the smallest positive solution of $Lh = 0$ in $D$ and $h = 1$ on $\partial D$. Define $X^h(t)$ to be the process $X(t)$ conditioned to hit $\partial D$. By Doob's $h$-transform theory, $X^h(t)$ is also a Markov diffusion and its generator $L^h$ is defined by $L^h f = (1/h)L(hf)$. Letting $\tau_D$ be the hitting time of $\partial D$, define the harmonic measure for $X^h(t)$ on $\partial D$ starting from $x \in D$ by $\mu^h_x(dy) = P_x^h(X^h(\tau_D) \in dy)$. Let $\{x_n\}^\infty_{n = 1} \subset D$ be a sequence satisfying $\lim_{n \rightarrow \infty}|x_n| = \infty$ for which $\mu^h_{x_n}$ converges weakly. Call two such sequences $\{x_n\}^\infty_{n = 1}$ and $\{x'_n\}^\infty_{n = 1}$ equivalent if $\lim_{n \rightarrow \infty}\mu^h_{x_n} = \lim_{n \rightarrow \infty} \mu^h_{x'_n}$. We call the set of equivalence classes thus generated the harmonic measure boundary at infinity for $L^h$. This boundary is independent of the particular exterior domain $D$. We prove that the harmonic measure boundary at infinity for $L^h$ coincides with the Martin boundary for $\tilde{L}$ on $R^d$, the formal adjoint of the operator $L$ on $R^d$. In the case that $L$ generates a reversible diffusion, the Martin boundaries of $L$ and $\tilde{L}$ coincide and hence the harmonic measure boundary of $L^h$ coincides with the Martin boundary for $L$ on $R^d$. A similar probabilistic description of the Martin boundary for $L$ on $R^d$ can be given in the nonreversible case. These results are then used to give explicit representations of the Martin boundaries of $L$ and $\tilde{L}$ for several classes of diffusion processes.

Consider a convex domain $B$ in $\mathbb{R}^n$ and denote by $p(t, x, y)$ the transition probability density of Brownian motion in $B$ killed at the boundary of $B$. The main result in this paper, in particular, shows that the function $s \ln s^np(s^2, x, y), (s, x, y) \in \mathbb{R}_+ \times B^2$, is concave.

We consider Brownian motion evolving among killing traps. We develop a technique of "enlargement of obstacles." This technique allows us to replace given trap configurations by configurations of enlarged traps, when deriving upper estimates on the probability that Brownian motion survives. Applied in a context of random obstacles, this reduces the complexity of the description for the environment seen by Brownian motion. We apply the method to the case where traps are distributed according to a fairly general Gibbs measure and obtain a result in the spirit of Donsker-Varadhan's theorem on Wiener sausage asymptotics.

We consider one-dimensional stochastic differential equations of the Stratonovich type: $dX_t = \sum_i\sigma_i(t, w, X_t)\circ dZ^i_t + \sum_k h_k(t, w, X_t)dA^k_t,$ where $Z^i$ are continuous semimartingales, and $A^k$ are continuous finite variation processes. We extend the definition of the Fisk-Stratonovich integral for a large class of coefficients $\sigma_i$, and under suitable conditions we prove existence and uniqueness for that equation.

In this paper we consider stochastic differential equations with reflecting boundary conditions for domains that might have corners and for which the allowed directions of reflection at a point on the boundary of the domain are possibly oblique. The main results are strong existence and uniqueness for solutions of such equations. A key ingredient is a family of relatively regular functions appropriate to the given domain and directions of reflection. Two cases are treated in the paper. In the first case the direction of reflection is single valued and varies smoothly, and the main new feature is that the boundary of the domain may be nonsmooth. In the second case the domain is taken to be the intersection of a finite number of domains with relatively smooth boundary, and at the resulting corner points more than one oblique direction is allowed.