Under the infinitely many sites mutation model, the mutational history of a sample of DNA sequences can be described by a unique gene tree. We show how to find the conditional distribution of the ages of the mutations and the time to the most recent common ancestor of the sample, given this gene tree. Explicit expressions for such distributions seem impossible to find for the sample sizes of interest in practice. We resort to a Monte Carlo method to approximate these distributions. We use this method to study the effects of variable population size and variable mutation rates, the distribution of the time to the most recent common ancestor of the population and the distribution of other functionals of the underlying coalescent process, conditional on the sample gene tree.

We prove turbulent diffusion theorems for Markovian velocity fields which either are mixing in time or have stationary vector potentials.

We describe the long-time behavior of the nonlinear pendulum perturbed by a small noise. To derive this asymptotics, one has to consider diffusion processes on the graph corresponding to the Hamiltonian of the pendulum.

In this paper, we give a pathwise development of stochastic integrals with respect to iterated Brownian motion. We also provide a detailed analysis of the variations of iterated Brownian motion. These variations are linked to Brownian motion in random scenery and iterated Brownian motion itself.

We consider a quenched-disordered heteropolymer, consisting of hydrophobic and hydrophylic monomers, in the vicinity of an oil-water interface. The heteropolymer is modeled by a directed simple random walk$(i, S_i)_{i\epsilon\mathbb{N}}$ on $\mathbb{N} \times \mathbb{Z}$ with an interaction given by the Hamiltonians $H_n^{\omega}(S) = \lambda \Sigma_{i=1}^n(\omega_i + h)\text{sign}(S_i)(n \epsilon \mathbb{N})$. Here, $\lambda$ and h are parameters and $(\omega_i)_{i\epsilon\mathbb{N}}$ are i.i.d. $\pm1$-valued random variables. The sign $(S_i) = \pm1$ indicates whether the ith monomer is above or below the interface, the $\omega_i = \pm1$ indicates whether the ith monomer is hydrophobic or hydrophylic. It was shown by Bolthausen and den Hollander that the free energy exhibits a localization-delocalization phase transition at a curve in the $(\lambda, h)$-plane.

In the present paper we show that the free-energy localization concept is equivalent to pathwise localization. In particular, we prove that free-energy localization implies exponential tightness of the polymer excursions away from the interface, strictly positive density of intersections with the interface and convergence of ergodic averages along the polymer. We include an argument due to G. Giacomin, showing that free-energy delocalization implies that there is pathwise delocalization in a certain weak sense.

Existence of solutions for stochastic Euler equations is proved for the two-dimensional case. The laws of solutions of stochastic Navier-Stokes equations are shown to be relatively compact and all limit points (as the viscosity converges to zero) are laws of solutions to stochastic Euler equations.

We prove a large deviations principle for the Young measures of a stochastic homogenization model of Poissonian biphased material.

Linear reaction-hyperbolic systems of partial differential equations in one space dimension arise in the study of the physiological process by which materials are transported in nerve cell axons. Probabilistic methods are developed to derive a closed form approximate solution for an initial-boundary value problem of such a system. The approximate solution obtained is a translating solution of a heat equation. An estimate is proved giving the deviation of this approximate traveling wave solution from the exact solution.

Mean-square approximations, which ensure boundedness of both time and space increments, are constructed for stochastic differential equations in a bounded domain. The proposed algorithms are based on a space-time discretization using a random walk over boundaries of small space-time parallelepipeds. To realize the algorithms, exact distributions for exit points of the space-time Brownian motion from a space-time parallelepiped are given. Convergence theorems are stated for the proposed algorithms. A method of approximate searching for exit points of the space-time diffusion from the bounded domain is constructed. Results of several numerical tests are presented.

Let $\mathbf{X} = {X_n: n = 1, 2,\dots}$ be a discrete valued stationary ergodic process distributed according to probability P. Let $\mathbf{Z}_1^n = {Z_1, Z_2,\dots, Z_n}$ be an independent realization of an n-block drawn with the same probability as X. We consider the waiting time $W_n$ defined as the first time the n-block $\mathbf{Z}_1^n$ appears in X. There are many recent results concerning this waiting time that demonstrate asymptotic properties of this random variable. In this paper, we prove that for all n the random variable $W_nP(Z_1^n)$ is approximately distributed as an exponential random variable with mean 1. We use a Poisson heuristic to provide a very simple intuition for this result, which is then formalized using the Chen-Stein method. We then rederive, with remarkable brevity, most of the known asymptotic results concerning $W_n$ and prove others as well. We further establish the surprising fact that for many sources $W_nP(\mathbf{Z}_1^n)$ is exp(1) even if the probability law for Z is not the same as that of X. We also consider the d-dimensional analog of the waiting time and prove a similar result in that setting. Nearly identical results are then derived for the recurrence time $R_n$ defined as the first time the initial N-block $\mathbf{X}_1^n$ reappears in X.

We conclude by developing applications of these results to provide concise solutions to problems that stem from the analysis of the Lempel-Ziv data compression algorithm. We also consider possible applications to DNA sequence analysis.

For the stable moving average process $$X^t = \int_{-\infty}^{\infty} f(t + x)M(dx), t = 1, 2,\dots,$$ we find the weak limit of its sample autocorrelation function as the sample size n increases to $\infty$. It turns out that, as a rule, this limit is random! This shows how dangerous it is to rely on sample correlation as a model fitting tool in the heavy tailed case. We discuss for what functions f this limit is nonrandom for all (or only some--this can be the case, too!) lags.

Fluid models have become a standard tool for demonstrating stability for queueing networks. It is presently not known, however, when the stability of a fluid model follows from that of the corresponding queueing network. We present an example of a queueing network where such stability does not, in fact, follow. This example also shows that the behavior of the fluid limits and the fluid model solutions for the same queueing network can differ considerably from one another.

We extend the results of Vvedenskaya, Dobrushin and Karpelevich to Jackson networks. Each node $j, 1 \leq j \leq J$ of the network consists of N identical channels, each with an infinite buffer and a single server with service rate $\mu_j$. The network is fed by a family of independent Poisson flows of rates $N\lambda_1,\dots, N\lambda_J$ arriving at the corresponding nodes. After being served at node j, a task jumps to node k with probability $p_{jk}$ and leaves the network with probability $p_j^* = 1 - \Sigma_k p_{jk}$. Upon arrival at any node, a task selects m of the N channels there at random and joins the one with the shortest queue. The state of the network at time $t \geq 0$ may be described by the vector $\underline{\mathbf{r}}(t) = {r_j(n, t), 1 \leq j \leq J, n \epsilon \mathbb{Z}_+}$, where $r_j(n, t)$ is the proportion of channels at node j with queue length at least n at time t. We analyze the limit $N \rightarrow \infty$. We show that, under a standard nonoverload condition, the limiting invariant distribution (ID) of the process $\underline{\mathbf{r}}$ is concentrated at a single point, and the process itself asymptotically approaches a single trajectory. This trajectory is identified with the solution to a countably infinite system of ODE's, whose fixed point corresponds to the limiting ID. Under the limiting ID, the tail of the distribution of queue-lengths decays superexponentially, rather than exponentially as in the case of standard Jackson networks--hence the term "fast networks" in the title of the paper.

An optimal investment policy model for the long term growth of expected utility of wealth is considered. The utility function is HARA with exponent $-\infty < \gamma < 1$. The problem can be reformulated as an infinite time horizon, risk sensitive control problem. Then the dynamic programming equations for different HARA exponents and different policy constraints are studied. We obtain some estimates for the solution of each equation. This can be used to derive an optimal policy with some interesting properties.

The paper studies the problem of maximizing the expected utility of terminal wealth in the framework of a general incomplete semimartingale model of a financial market. We show that the necessary and sufficient condition on a utility function for the validity of several key assertions of the theory to hold true is the requirement that the asymptotic elasticity of the utility function is strictly less than 1.