We develop some topics about Brownian motion with a particular emphasis on the study of principal values of Brownian local times. We show some links between principal values and Doob’s $h$-transforms of Brownian motion, for nonpositive harmonic functions $h$. We also give a survey and complement some martingale approaches to Ray–Knight theorems for local times.

We introduce a random recursive fractal based on the Sierpinski gasket and construct a diffusion upon the fractal via a Dirichlet form. This form and its symmetrizing measure are determined by the electrical resistance of the fractal. The effective resistance provides a metric with which to discuss the properties of the fractal and the diffusion. The main result is to obtain uniform upper and lower bounds for the transition density of the Brownian motion on the fractal in terms of this metric. The bounds are not tight as there are logarithmic corrections due to the randomness in the environment, and the behavior of the shortest paths in the effective resistance metric is not well understood. The results are deduced from the study of a suitable general branching process.

We study a family of differential operators ${L^{\alpha}, \alpha \geq 0}$ in the unit ball $D$ of $C^n$ with $n \geq 2$ that generalize the classical Laplacian, $\alpha = 0$, and the conformal Laplacian, $\alpha = 1/2$ (that is, the Laplace-Beltrami operator for Bergman metric in $D$). Using the diffusion processes associated with these (degenerate) differential operators, the boundary behavior of $L^{\alpha}$-harmonic functions is studied in a unified way for $0 \leq \alpha \leq 1/2$. More specifically, we show that a bounded $L^{\alpha}$-harmonic function in $D$ has boundary limits in approaching regions at almost every boundary point and the boundary approaching region increases from the Stolz cone to the Korányi admissible region as $\alpha$ runs from 0 to 1/2. A local version for this Fatou-type result is also established.

Suppose ${\Im_t}$ is the filtration induced by a Wiener process $W$ in $R^d$, $\tau$ is a finite ${\Im_t}$ stopping time (terminal time), $\xi$ is an ${\Im_{\tau}}$-measurable random variable in $R^k$ (terminal value) and $f(\cdot, y, z)$ is a coefficient process, depending on $y \in R^k$ and $z \in L(R^d, R^k)$, satisfying $(y - \tilde{y})[f(s, y, z) - f(s, \tilde{y}, z)] \leq - a|y - \tilde{y}|^2$ ($f$ need not be Lipschitz in $y$), and $|f(s, y, z) - f(s, y, \tilde{z})| \leq b||z - \tilde{z}||$, for some real $a$ and $b$, plus other mild conditions. We identify a Hilbert space, depending on $\tau$ and on the number $\gamma \equiv b^2 - 2a$, in which there exists a unique pair of adapted processes $(Y, Z)$ satisfying the stochastic differential equation $$dY(s) = 1_{{s \leq \tau}} [Z(s) dW(s) - f(s, Y(s), Z(s)) ds]$$ with the given terminal condition $Y(\tau) - \xi$, provided a certain integrability condition holds. This result is applied to construct a continuous viscosity solution to the Dirichlet problem for a class of semilinear elliptic PDE’s.

This paper is concerned with the intrinsic metrics of the two main classes of superprocesses. For the Fleming-Viot process, we identify it as the Bhattacharya distance, and for Dawson-Watanabe processes, we find the Kakutani-Hellinger metric. The corresponding geometries are studied in some detail. In particular, representation formulas for geodesics and arc length functionals are obtained. The relations between the two metrics yield a geometric interpretation of the identification of the Fleming-Viot process as a Dawson-Watanabe superprocess conditioned to have total mass 1. As an application, a functional limit theorem for super-Brownian motion conditioned on local extinction is proved.

We describe a coarse graining method, which provides lower bounds on the principal Dirichlet eigenvalue of the Laplacian in regions receiving small obstacles, and sharpens the previous method of enlargement of obstacles. Based on a quantitative Wiener criterion, one replaces the actual obstacles by obstacles of a much larger size. Controls on the shift of principal eigenvalues and capacity estimates on the locus where the Wiener criterion breaks down are derived. The results are written in a self-contained fashion.

We give several examples and examine case studies of linear stochastic functional differential equations. The examples fall into two broad classes: regular and singular, according to whether an underlying stochastic semi-flow exists or not. In the singular case, we obtain upper and lower bounds on the maximal exponential growth rate $\overline{\lambda}_1 (\sigma)$ of the trajectories expressed in terms of the noise variance ƒÐ . Roughly speaking we show that for small $\sigma$, $\overline{\lambda}_1 (\sigma)$ behaves like $-\sigma^2 /2$, while for large $\sigma$, it grows like $\log \sigma$. In the regular case, it is shown that a discrete Oseledec spectrum exists, and upper estimates on the top exponent $\lambda_1$ are provided. These estimates are sharp in the sense that they reduce to known estimates in the deterministic or nondelay cases.

We obtain a formula for Liapounov exponents of stochastic flows generated by stochastic differential equations on compact manifolds. As an application, the exponents of a class of stochastic flows on spheres are determined explicitly.

A combinatorial definition of multiple stochastic integrals is given in the setting of random measures. It is shown that some properties of such stochastic integrals, formerly known to hold in special cases, are instances of combinatorial identities on the lattice of partitions of a set. The notion of stochastic sequences of binomial type is introduced as a generalization of special polynomial sequences occuring in stochastic integration, such as Hermite, Poisson–Charlier and Kravchuk polynomials. It is shown that identities for such polynomial sets have a common origin.

We study the question of whether the quasilocality (continuity, almost Markovianness) property of Gibbs measures remains valid under a projection on a sub-$\sigma$-algebra. Our method is based on the construction of global specifications, whose projections yield local specifications for the projected measures. For Gibbs measures compatible with monotonicity preserving local specifications, we show that the set of configurations where quasilocality is lost is an event of the tail field. This set is shown to be empty whenever a strong uniqueness property is satisfied, and of measure zero when the original specification admits a single Gibbs measure. Moreover, we provide a criterion for nonquasilocality (based on a quantity related to the surface tension). We apply these results to projections of the extremal measures of the Ising model. In particular, our nonquasilocality criterion allows us to extend and make more complete previous studies of projections to a sublattice of one less dimension (Schonmann example).

For spin systems with random, finite-range interactions, we define an analog of the usual weak mixing property, which we call “weak mixing in expectation” (WME). This property implies (almost sure) uniqueness of the Gibbs measure.

We concentrate on the two-dimensional case, for which we present finite-volume conditions which are sufficient for WME. We also show the reverse: if the system is WME, then the condition is satisfied for some (sufficiently large) volume. Simultaneously, we obtain an extension (to random interactions) of the result by Martinelli, Olivieri and Schonmann that weak mixing implies strong mixing.

Our method is based on a rescaled version of the disagreement percolation approach of van den Berg and Maes, combined with ideas and techniques of Gielis and Maes, and Martinelli, Olivieri and Schonmann. However, apart from some general results on coupling, stated in Section 2, this paper is self-contained.

Consider the directed process $(i, S_i)$ where the second component is simple random walk on $\mathbb{Z} (S_0 = 0)$. Define a transformed path measure by weighting each $n$-step path with a factor $\exp [\lambda \sum_{1 \leq i \leq n}(\omega_i + h)\sign (S_i)]$. Here, $(\omega_i)_{i \geq 1}$ is an i.i.d. sequence of random variables taking values $\pm 1$ with probability 1/2 (acting as a random medium) , while $\lambda \in [0, \infty)$ and $h \in [0, 1)$ are parameters. The weight factor has a tendency to pull the path towards the horizontal, because it favors the combinations $S_i > 0, \omega_i = +1$ and $S_i < 0, \omega_i = -1$. The transformed path measure describes a heteropolymer, consisting of hydrophylic and hydrophobic monomers, near an oil-water interface.

We study the free energy of this model as $n \to \infty$ and show that there is a critical curve $\lambda \to h_c (\lambda)$ where a phase transition occurs between localized and delocalized behavior (in the vertical direction). We derive several properties of this curve, in particular, its behavior for $\lambda \downarrow 0$. To obtain this behavior, we prove that as $\lambda, h \downarrow 0$ the free energy scales to its Brownian motion analogue.

We study the asymptotic behavior of symmetric spin glass dynamics in the Sherrington–Kirkpatrick model as proposed by Sompolinsky–Zippelius. We prove that the averaged law of the empirical measure on the path space of these dynamics satisfies a large deviation upper bound in the high temperature regime. We study the rate function which governs this large deviation upper bound and prove that it achieves its minimum value at a unique probability measure $Q$ which is not Markovian. We deduce an averaged and a quenched law of large numbers. We then study the evolution of the Gibbs measure of a spin glass under Sompolinsky–Zippelius dynamics. We also prove a large deviation upper bound for the law of the empirical measure and describe the asymptotic behavior of a spin on path space under this dynamic in the high temperature regime.

We study dependent bond percolation on the homogeneous tree $T_n$ of order $n \geq 2$ under the assumption of automorphism invariance. Excluding a trivial case, we find that the number of infinite clusters a.s. is either 0 or $\infty$. Furthermore, each infinite cluster a.s. has either 1, 2 or infinitely many topological ends, and infinite clusters with infinitely many topological ends have a.s. a branching number greater than 1. We also show that if the marginal probability that a single edge is open is at least $2/(n + 1)$, then the existence of infinite clusters has to have positive probability. Several concrete examples are considered.

A composition structure is a sequence of consistent probability distributions for compositions (ordered partitions) of $n = 1, 2, \dots$. Any composition structure can be associated with an exchangeable random composition of the set of natural numbers. Following Donnelly and Joyce, we study the problem of characterizing a generic composition structure as a convex mixture of the "extreme" ones. We topologize the family $\mathscr{U}$ of open subsets of [0, 1] so that $\mathscr{U}$ becomes compact and show that $\mathscr{U}$ is homeomorphic to the set of extreme composition structures. The general composition struc-ture is related to a random element of $\mathscr{U}$ via a construction introduced by J. Pitman.

This paper quantifies the degree to which exponential bounds can be used to approximate tail probabilities of partial sums of arbitrary i.i.d. random variables. The introduction of a single truncation allows the usual exponential upper bound to apply usefully whenever the summands are arbitrary i.i.d. random variables. More specifically, let n be a fixed natural number and let $Z, Z_1, Z_2, \dots, Z_n$ be arbitrary i.i.d. random variables. We construct a function $F_{Z, n} (a)$, derived from the probability of occurrence of one or more ‘‘large’’ summands plus an upper bound of exponential type, such that for some constant $C_* > 0$ (independent of $Z, n$ and $a$) and all real $a$,

$$C_* F_{Z,n}^2 (a) \leq P(\sum_{j=1}^n Z_j \geq na) \leq 2F_{Z,n} (a).$$

Furthermore, examples show that the upper and lower bounds are achievable.

Let $X_1, Y_1, Y_2, \dots, X_n, Y_n$ be independent nonnegative rv’s and let $\{b_{ij}\}_{1 \leq i, j \leq n}$ be an array of nonnegative constants. We present a method of obtaining the order of magnitude of

$$E \Phi (\sum_{1 \leq i, j \leq n} b_{ij} X_i Y_j),$$

for any such ${X_i}, {Y_j}$ and ${b_{ij}}$ and any nondecreasing function $\Phi$ on $[0, \infty)$ with $\Phi (0) = 0$ and satisfying a $\Delta_2$ growth condition. Furthermore, this technique is extended to provide the order of magnitude of

$$E \Phi (\sum_{1 \leq i, j \leq n} f_{ij} (X_i Y_j)),$$

where ${f_{ij} (x, y)}_{1 \leq i, j \leq n}$ is any array of nonnegative functions.

For arbitrary functions ${g_{ij} (x, y)}_{1 \leq i \neq j \leq n}$, the aforementioned approximation enables us to identify the order of magnitude of

$$E \Phi (|\sum_{1 \leq i \not= j \leq n} g_{ij} (X_i Y_j)|),$$

whenever decoupling results and Khintchine-type inequalities apply, such as $\Phi$ is convex, $\mathscr{L} (g_{ij}(X_i, X_j)) = \mathscr{L}(g_{ij}(X_j, X_i))$ and $Eg_{ij}(X_i, x) \equiv 0$ for all $x$ in the range of $X_j$.

For the sum $S = \sum X_i$ of a sequence $(X_i)$ of independent symmetric (or nonnegative) random variables, we give lower and upper estimates of moments of $S$. The estimates are exact, up to some universal constants, and extend the previous results for particular types of variables $X_i$.

Let $X, X_i, i \in \mathbb{N}$, be independent, identically distributed random variables. It is shown that the Student $t$-statistic based upon the sample ${X_i}_{i=1}^n$ is asymptotically $N(0, 1)$ if and only if $X$ is in the domain of attraction of the normal law. It is also shown that, for any $X$, if the self-normalized sums $U_n := \sum_{i=1}^n X_i/(\sum_{i=1}^n X_i^2)^{1/2}, n \in \mathbb{N}$, are stochastically bounded then they are uniformly subgaussian that is, $\sup_n \mathbb{E} \exp (\lambda U_n^2) < \infty$ for some $\lambda > 0$.

The set $\mathscr{P}(S)$ of probability measures on the Borel sigma field of every closed subset $S$ of $\mathbb{R}^k$ is shown to be complete under a Kolmogorov type metric. An assertion in the 1988 article by Bhattacharya and Lee that this completeness holds if $S$ is merely topologically complete is false.