The lattice gas model in equilibrium is considered. We give a lower bound of the density-density time correlation for large time, which involves the bulk diffusion matrix in a physically natural way.

Consider two exchangeable sequences $(X_k)_{k \in N}$ and $(\hat{X}_k)_{k \in N}$ with the property that $S_n \equiv \sum_{k=1}^n X_k$ and $\hat{S}_n \equiv \sum_{k=1}^n \hat{X}_k$ have the same distribution for all $n \in N$. David Aldous posed the following question. Does this imply that the two exchangeable sequences have the same joint distributions? We give an example that shows the answer to Aldous' question is, in general, in the negative. On the other hand, we show that the joint distributions of an exchangeable sequence can be recovered from the distributions of its partial sums if the sequence is a countable mixture of i.i.d. sequences that are either nonnegative or have finite moment generating functions in some common neighbourhood of zero.

Rogers-Pitman have shown that the sum of the absolute value of $B^{(\mu)}$, Brownian motion with constant drift $\mu$, and its local time $L^{(\mu)}$ is a diffusion $R^{(\mu)}$. We exploit the intertwining relation between $B^{(\mu)}$ and $R^{(\mu)}$ to show that the same addition operation performed on a one-parameter family of diffusions ${X^{(\alpha,\mu)}}_{\alpha\in{\mathbf R}_+}$ yields the same diffusion $R^{(\mu)}$. Recently we obtained an exponential analogue of the Rogers-Pitman result. Here we exploit again the corresponding intertwining relationship to yield a one-parameter family extension of our result.

The law of a random tree constructed within a Brownian excursion is calculated conditional on knowing the occupation measure of the excursion. In previous work David Aldous has used random walk approximations to obtain this result. Here it is deduced from Le Gall's description of the tree in the unconditioned excursion.

We construct a Brownian path conditioned on its minimum value over a fixed time interval by a simple transformation of a Brownian bridge.

Noise sensitivity of functions on the leaves of a binary tree is studied, and a hypercontractive inequality is obtained. We deduce that the spider walk is not noise stable.

The purpose of this note is to make available a reasonably complete and straightforward proof of Strassen's theorem on stochastic domination, and to draw attention to the original paper. We also point out that the maximal possible value of $P(Z = Z')$ is actually not reduced by the requirement $Z \leq Z'$. Here, $Z,Z'$ are stochastic elements that Strassen's theorem states exist under a stochastic domination condition. The consequence of that observation to stochastically monotone Markov chains is pointed out. Usually the theorem is formulated with the assumption that $\leq$ is a partial ordering; the proof reveals that a pre-ordering suffices.

Recently, Kalvin M. Jansons derived in an elegant way the Laplace transform of the time spent by an excursion above a given level $a \gt 0$. This result can also be derived from previous work of the author on the occupation time of the excursion in the interval $(a,a+b]$, by sending $b \to \infty$. Several alternative derivations areincluded.

Fleming-Viot processes incorporating mutation and selection are considered. It is well-known that if the mutation factor is of uniform type, the process has a reversible stationary distribution, and it has been an open problem to characterize the class of the processes that have reversible stationary distributions. This paper proves that if a Fleming-Viot process has a reversible stationary distribution, then the associated mutation operator is of uniform type.

We study a class of Borel probability measures, called correlation measures. Our results are of two types: first, we give explicit constructions of non-trivial correlation measures; second, we examine some of the properties of the set of correlation measures. In particular, we show that this set of measures has a convexity property. Our work is related to the so-called Gaussian correlation conjecture.

In this note, we study the Gaussian fluctuations of the edge occupation measure for a reversible Markov chain and give a nice description of the covariance matrix. Then we give some large deviations results concerning this occupation measure.

We consider an isotropic Brownian flow on $R^d$ for $d\geq 2$ with a positive Lyapunov exponent, and show that any nontrivial connected set almost surely contains points whose distance from the origin under the flow grows linearly with time. The speed is bounded below by a fixed constant, which may be computed from the covariance tensor of the flow. This complements earlier work, which showed that stochastic flows with bounded local characteristics and zero drift cannot grow at a linear rate faster than linear.

Let $T_1$ denote the first passage time to 1 of a standard Brownian motion. It is well known that as $\lambda$ goes to infinity, $P\{ T_1 \gt \lambda \}$ goes to zero at rate $c \lambda^{-1/2}$, where $c$ equals $(2/ \pi)^{1/2}$. The goal of this note is to establish a quantitative, infinite dimensional version of this result. Namely, we will prove the existence of positive and finite constants $K_1$ and $K_2$, such that for all $\lambda \gt e^e$, $$K_1 \lambda^{-1/2} \leq \text{Cap} \{ T_1 \gt \lambda\} \leq K_2 \lambda^{-1/2} \log^3(\lambda) \cdot \log\log(\lambda),$$ where `$\log$' denotes the natural logarithm, and $\text{Cap}$ is the Fukushima-Malliavin capacity on the space of continuous functions.

We present a Gaussian correlation inequality which is closely related to a result of Schechtman, Schlumprecht and Zinn (1998) on the well-known Gaussian correlation conjecture. The usefulness of the inequality is demonstrated by several important applications to the estimates of small ball probability.