The equation $u_t = u_{xx} + u(1 - u)$ is the simplest reaction-diffusion equation. Introduction of a small parameter allows construction of geometric optics approximations for the solutions of such equations; these solutions are approximated by step-functions with the values 0 and 1. The region where the solution is close to 1 propagates according to the Huygens principle for the corresponding velocity field $v(x, e)$ which is calculated via the equation. New effects may emerge, such as stops and jumps of the wave front. The Feynman-Kac formula implies that the solutions of certain Cauchy problems obey some integral equations in the space of trajectories of the corresponding Markov processes. Examination of this equation requires the study of Laplace-type asymptotics for functional integrals. These asymptotics are defined by large deviations for the corresponding family of processes and are expressed through action functionals.

The Donsker-Varadhan $I$-Function which measures the asymptotic rate of decay of the distribution of the occupation measure of a Markov process is evaluated for diffusion processes with boundaries.

Let $D$ be a connected, compact region in $R^d$. If $d = 1$, then for each nice probability measure $\mu$ on $D$ and diffusion coefficient $a$, there exists a unique drift such that $\mu$ is invariant for the resulting diffusion process with reflection at the boundary. For $d > 1$, there is no uniqueness. For each diffusion matrix $a$, reflection vector $J$, and nice probability measure $\mu$ on $D$, we classify the collection of drifts such that $\mu$ is invariant for the resulting diffusion process. We use the theory of the $I$-function and, in the course of things, answer a question about the $I$-function.

We give the asymptotic estimate for large $t$ of elementary solutions of one-dimensional generalized diffusion equations with regularly varying Green functions. As a corollary we obtain the precise asymptotic behavior of the semigroup $T_tf(x)$ for all $f \in L_1(dm)$ if the speed measure function $m(x)$ is regularly varying as $x \rightarrow \pm \infty$.

The classical results on the instability of the solutions of stochastic differential equations are extended in two directions: the coefficients are allowed to depend on the paths of the solutions, and arbitrary semimartingales (not simply continuous ones) are allowed as differentials. This extends the results of Wong and Zakai, McShane, Nakao and Yamato, etc.

Suppose given a smooth, compact planar region $S$ and a smooth inward pointing vector field on $\partial S$. It is known that there is a diffusion process $Z$ which behaves like standard Brownian motion inside $S$ and reflects instantaneously at the boundary in the direction specified by the vector field. It is also known $Z$ has a stationary distribution $p$. We find a simple, general explicit formula for $p$ in terms of the conformal map of $S$ onto the upper half plane. We also show that this formula remains valid when $S$ is a bounded polygon and the vector field is constant on each side. This polygonal case arises as the heavy traffic diffusion approximation for certain two-dimensional queueing and storage processes.

The object of study in this paper is reflected Brownian motion in a two-dimensional wedge with constant direction of reflection on each side of the wedge. The following questions are considered. Is the process recurrent? If it is recurrent, what is its invariant measure? Let $\xi$ be the angle of the wedge $(0 < \xi < 2\pi)$ and let $\theta_1$ and $\theta_2$ be the angles of reflection on the two sides of the wedge, measured from the inward normals towards the directions of reflection, with positive angles being toward the corner $(-\pi/2 < \theta_1, \theta_2 < \pi/2)$. Set $\alpha = (\theta_1 + \theta_2)/\xi$. Varadhan and Williams (1985) have shown that the process exists and is unique, in the sense that it solves a certain submartingale problem, when $\alpha < 2$. It is shown here that if $\alpha < 0$, the process is transient (to infinity). If $0 \leq \alpha < 2$, the process is shown to be (finely) recurrent and to have a unique (up to a scalar multiple) $\sigma$-finite invariant measure. It is further proved that the density for this invariant measure is given in polar coordinates by $p(r, \theta) = r^{-\alpha}\cos(\alpha\theta - \theta_1)$.

It is known that if $B_t$ is a standard Wiener process then $\sup_t\lim \inf_{h\rightarrow 0+}(B_{t + h} - B_t)h^{-1/2} = 1 a.s.$ Here this is sharpened to $P(\exists t: \lim \inf_{h\rightarrow 0+}(B_{t+h} - B_t)h^{-1/2} = 1) = 1$, and $P(\exists t: B_{t + h} - B_t \geq h^{1/2}\forall h \in (0, \alpha)$ for some $\alpha > 0) = 0$. A number of other theorems of the same flavor are proved. Our results for the critical case for slow points are not as complete as the above.

Let $\eta(t)$ be stationary, Gaussian, and suppose one wants to predict the future upcrossings of a certain level $u$. The paper investigates criteria for a good level crossing predictor, and restates in simple form a result by de Mare on optimal prediction. Write $\hat{\eta}_t(t + h)$ for the mean square predictor of $\eta(t + h)$ at time $t$, and let $\hat{\xi}_t(t + h)$ be the conditional expectation of $\eta'(t + h)$ given observed data and given $\eta(t + h) = u$. It is shown that an alarm which predicts an upcrossing at $t + h$ if $\hat{\eta}_t(t + h)$ differs from $u$ by a quantity that is a certain function of $\hat{\zeta}_t(t + h)$ is optimal in the sense that it maximizes the detection probability for a fixed total alarm time. Explicit formulas are given for the upcrossing risks at alarm, detection probability, and total alarm time. In an example the optimal alarm is compared to a naive alarm which predicts an upcrossing if $\hat{\eta}_t(t + h)$ differs from $u$ by a fixed proportion of the residual standard deviation. The optimal alarm locates the upcrossings more precisely and at an earlier stage than the naive alarm, which has a tendency to give late alarms.

If $X$ takes values in a Banach space $B$ and is in the domain of normal attraction of a Gaussian law on $B$ with $EX = 0, E(\|X\|^2/L_2\|X\|) < \infty$, then it is known that $X$ satisfies the compact law of the iterated logarithm as described in Goodman, Kuelbs and Zinn [9], Theorem 4.1. In this paper the analogous result is demonstrated when $X$ is merely in the domain of attraction of a Gaussian law. The functional LIL is also obtained in this setting. These results refine Corollary 7 of Kuelbs and Zinn [22], as well as various functional LILs.

In this paper we consider weak convergence of processes indexed by a collection $\mathscr{A}$ of subsets of $I^d$. As a suitable sample space for such processes, we introduce the space $\mathscr{D}(\mathscr{A})$ of set functions that are outer continuous with inner limits. A metric is defined for $\mathscr{D}(\mathscr{A})$ in terms of the graphs of its elements and then we give a sufficient condition for a subset of $\mathscr{D}(\mathscr{A})$ to be compact in this topology. This framework is then used to provide a criterion for probability measures on $\mathscr{D}(\mathscr{A})$ to be tight. As an application, we prove a central limit theorem for partial-sum processes indexed by a family of sets, $\mathscr{A}$, when the underlying random variables are in the domain of normal attraction of a stable law. If $\alpha \in (1, 2)$ denotes the exponent of the limiting stable law, if $r$ denotes the coefficient of metric entropy of $\mathscr{A}$, and if $\mathscr{A}$ satisfies mild regularity conditions, we show that the partial-sum processes converge in law to a stable Levy process provided $r < (\alpha - 1)^{-1}$.

Necessary and sufficient conditions are given for the almost sure convergence of the quadratic form $\sum \sum f_{jk}M_jM_k$ where $(M_j)$ is a sequence of i.i.d. $p$-stable random variables. A connection is established between the convergence of the quadratic form and a radonifying property of the infinite matrix operator $(f_{kj})$.

A famous theorem of De Finetti (1931) shows that an exchangeable sequence of $\{0, 1\}$-valued random variables is a unique mixture of coin tossing processes. Many generalizations of this result have been found; Hewitt and Savage (1955) for example extended De Finetti's theorem to arbitrary compact state spaces (instead of just $\{0, 1\}$). Another type of question arises naturally in this context. How can mixtures of independent and identically distributed random sequences with certain specified (say normal, Poisson, or exponential) distributions be characterized among all exchangeable sequences? We present a general theorem from which the "abstract" theorem of Hewitt and Savage as well as many "concrete" results--as just mentioned--can be easily deduced. Our main tools are some rather recent results from harmonic analysis on abelian semigroups.

This paper deals with a modification of Galton-Watson processes allowing random migration in the following way: with a probability $p_n$(in the nth generation) one particle is eliminated and does not take part in further evolution, or with a probability $r_n$ takes place immigration of new particles according to a p.g.f. $G(s)$, and, finally, with a probability $q_n$ there is not any migration, $p_n + q_n + r_n = 1, n = 0, 1, 2, \cdots$. We investigate a critical case when the offspring mean is equal to one and $r_nG'(1) \equiv p_n \rightarrow 0$. Depending on the rate of this convergence we obtain different types of limit theorems.

The conditional independence relation for a triple of $\sigma$-algebras is investigated. For certain operations on this relation, necessary and sufficient conditions are derived such that these operations leave the relation invariant. Examples of such operations are the enlargement or reduction of the $\sigma$-algebras, and an absolute continuous change of measure. A projection operator for $\sigma$-algebras is defined and some of its properties are stated. The $\sigma$-algebraic realization problem is briefly discussed.

Given a two-parameter filtration $(\mathscr{F}_z)$ satisfying the conditional independence assumption (F4), we prove the existence of an optimal stopping point for adapted processes $(X_z)$ indexed by $\mathbb{N}^2$ or $\mathbb{R}^2_+$ which are of class $(D)$, and have regularity properties which generalize the usual one-parameter ones, and are expressed in terms of sequences of 1- and 2-stopping points.

An inequality due to Chernoff is generalized and a related Cramer-Rao type of inequality is studied.

A frequently occurring problem is to find a probability distribution lying within a set $\mathscr{E}$ which minimizes the $I$-divergence between it and a given distribution $R$. This is referred to as the $I$-projection of $R$ onto $\mathscr{E}$. Csiszar (1975) has shown that when $\mathscr{E} = \cap^t_1 \mathscr{E}_i$ is a finite intersection of closed, linear sets, a cyclic, iterative procedure which projects onto the individual $\mathscr{E}_i$ must converge to the desired $I$-projection on $\mathscr{E}$, provided the sample space is finite. Here we propose an iterative procedure, which requires only that the $\mathscr{E}_i$ be convex (and not necessarily linear), which under general conditions will converge to the desired $I$-projection of $R$ onto $\cap^t_1 \mathscr{E}_i$.

For the study of the paths of stochastic processes, the upcrossing-down-crossing arguments are available in the real-valued case but not in the vector-valued case. We use in this article stopping time methods and the Kadec-Klee renorming theorem to obtain regularity properties for Banach-lattice valued submartingales and Banach-valued pramarts.

Let $n k$-dimensional spheres, each of content $a_n$, be distributed within a $k$-dimensional cube according to density $f$. We derive necessary and sufficient conditions on $a_n$ in order that the probability that the cube is completely covered at least $\ell$ times by the spheres, tend to one as $n\rightarrow\infty$. (Here $\ell$ is an arbitrary positive integer.) In the special case $f\equiv$ const., we obtain upper and lower bounds of the same order of magnitude for the probability of incomplete coverage.

For $f$ analytic in the unit disc put $\nu(f) = \int^\tau_0 |f'(B(s))|^2 ds$ where $\tau$ is the exit time of Brownian motion $B(t)$ from the disc. We prove that $E\Phi(\tau) \leq E\Phi(\nu(f))$ for all $f$ satisfying $|f'(0)| = 1$ and a wide class of $\Phi$. In particular, we may take $\Phi(\lambda) = |\lambda|^P$ for $0 < p < \infty$.

If $\cos(\theta/2) < 1/\sqrt n$ then a.s. there are times $0 \leq s_1 < s_2$ such that the $n$-dimensional Brownian motion $Z(t)$ stays for all $t \in (s_1, s_2)$ in a cone with vertex $Z(s_1)$ and angle $\theta$. If $\cos(\theta/2) > 1/\sqrt n$ then the same event has probability 0.

Jump behavior of the first passage time processes for Brownian motion, its range and BES(3) are compared via their Poisson measures. Explicit results concerning Brownian motion up to a first passage time of its range are given.

The main purpose of this paper is to draw attention to a simple and useful general "continuity theorem" type result, from which a great deal of limit theorems follow as almost immediate consequences. As an example, we give a new very short and transparent proof of the recent result by H. Teicher and C. Hagwood (A multidimensional CLT for maxima of normed sums); in fact, a much more general assertion is proved here. Another application of the main result establishes a correspondence between the convergence of empirical and quantile processes. (A similar result holds for the renewal and partial sums processes.)

Let $\mu$ be a Radon probability measure on a type 2 Banach space $E$. The following Bochner's theorem is proved. For every continuous positive definite function $\phi(\phi(0) = 1)$ on $E$, there exists a Radon probability measure $\sigma_\phi$ on the measurable dual $H_0(\mu)$ of $(E, \mu)$ with the characteristic functional $\phi$ (in some restricted sense).

Let $(p_{ij})$ be the matrix of a recurrent Markov chain with stationary vector $\pi > 0,$ and let $\{J_i\}$ be a partition of the unit circle into sets $J_1, \cdots, J_n, m(J_i) = \pi_i$, where $m$ is Lebesgue measure. Suppose $f_t$ defines rotation through distance $t$. The conditions under which $p_{ij}$ can be written as $m(f_t(J_i) \cap J_j)/m(J_i)$ for all $i$ and $j$, where each $J_i$ is the union of at most $b(n)$ arcs, have recently been examined by Steve Alpern and Joel Cohen. Cohen conjectured that $b(n) = n - 1$, and proved $b(2) = 1$. Alpern proved that Cohen's conjecture was false for $n$ sufficiently large, and gave bounds for $b(n)$. We give a construction that shows that $b(3) = 2$, and prove that $b(n)$ is nondecreasing.

We show that for the transition kernels $(\mu_y)$ of a certain random walk in $\mathbb{R}^2$ and the Radon Transform in $\mathbb{R}^3$ there is no subset $K$ of positive Lebesgue-measure such that $(\mu_y)_{y\in K}$ is completely orthogonal.