We study a model of a polling system, that is, a collection of d queues with a single server that switches from queue to queue. The service time distribution and arrival rates change randomly every time a queue is emptied. This model is mapped to a mathematically equivalent model of a random walk with random choice of transition probabilities, a model which is of independent interest. All our results are obtained using methods from the constructive theory of Markov chains. We determine conditions for the existence of polynomial moments of hitting times for the random walk. An unusual phenomenon of thickness of the region of null recurrence for both the random walk and the queueing model is also proved.

We propose two models of the evolution of a pair of competing populations. Both are lattice based. The first is a compromise between fully spatial models, which do not appear amenable to analytic results, and interacting particle system models, which do not, at present, incorporate all of the competitive strategies that a population might adopt. The second is a simplification of the first, in which competition is only supposed to act within lattice sites and the total population size within each lattice point is a constant. In a special case, this second model is dual to a branching annihilating random walk. For each model, using a comparison with oriented percolation, we show that for certain parameter values, both populations will coexist for all time with positive probability. As a corollary, we deduce survival for all time of branching annihilating random walk for sufficiently large branching rates. We also present a number of conjectures relating to the rôle of space in the survival probabilities for the two populations.

We study dioecious (i.e., two-sex) branching particle system models, where there are two types of particles, modeling the male and female populations, and where birth of new particles requires the presence of both male and female particles. We show that stationary distributions of various dioecious branching particle models are nontrivial under certain conditions, for example, when there is sufficiently fast stirring.

We propose a general method to study dependent data in a binary tree, where an individual in one generation gives rise to two different offspring, one of type 0 and one of type 1, in the next generation. For any specific characteristic of these individuals, we assume that the characteristic is stochastic and depends on its ancestors’ only through the mother’s characteristic. The dependency structure may be described by a transition probability P(x, dy dz) which gives the probability that the pair of daughters’ characteristics is around (y, z), given that the mother’s characteristic is x. Note that y, the characteristic of the daughter of type 0, and z, that of the daughter of type 1, may be conditionally dependent given x, and their respective conditional distributions may differ. We then speak of bifurcating Markov chains.

We derive laws of large numbers and central limit theorems for such stochastic processes. We then apply these results to detect cellular aging in Escherichia Coli, using the data of Stewart et al. and a bifurcating autoregressive model.

Several results of large deviations are obtained for distributions that are associated with the Poisson–Dirichlet distribution and the Ewens sampling formula when the parameter θ approaches infinity. The motivation for these results comes from a desire of understanding the exact meaning of θ going to infinity. In terms of the law of large numbers and the central limit theorem, the limiting procedure of θ going to infinity in a Poisson–Dirichlet distribution corresponds to a finite allele model where the mutation rate per individual is fixed and the number of alleles going to infinity. We call this the finite allele approximation. The first main result of this article is concerned with the relation between this finite allele approximation and the Poisson–Dirichlet distribution in terms of large deviations. Large θ can also be viewed as a limiting procedure of the effective population size going to infinity. In the second result a comparison is done between the sample size and the effective population size based on the Ewens sampling formula.

It is shown that the method of exchangeable pairs introduced by Stein [Approximate Computation of Expectations (1986) IMS, Hayward, CA] for normal approximation can effectively be used for translated Poisson approximation. Introducing an additional smoothness condition, one can obtain approximation results in total variation and also in a local limit metric. The result is applied, in particular, to the anti-voter model on finite graphs as analyzed by Rinott and Rotar [Ann. Appl. Probab. 7 (1997) 1080–1105], obtaining the same rate of convergence, but now for a stronger metric.

Let L be a multidimensional Lévy process under P in its own filtration. The f^q-minimal martingale measure Q_q is defined as that equivalent local martingale measure for $\mathcal {E}(L)$ which minimizes the f^q-divergence E[(dQ/dP)^q] for fixed q∈(−∞, 0)∪(1, ∞). We give necessary and sufficient conditions for the existence of Q_q and an explicit formula for its density. For q=2, we relate the sufficient conditions to the structure condition and discuss when the former are also necessary. Moreover, we show that Q_q converges for q↘1 in entropy to the minimal entropy martingale measure.

The two-type Richardson model describes the growth of two competing infections on ℤ^d and the main question is whether both infection types can simultaneously grow to occupy infinite parts of ℤ^d. For bounded initial configurations, this has been thoroughly studied. In this paper, an unbounded initial configuration consisting of points x=(x₁, …, x_d) in the hyperplane $\mathcal{H}=\{x\in\mathbb{Z}^{d}:x_{1}=0\}$ is considered. It is shown that, starting from a configuration where all points in $\mathcal{H}\backslash\{\mathbf{0}\}$ are type 1 infected and the origin 0 is type 2 infected, there is a positive probability for the type 2 infection to grow unboundedly if and only if it has a strictly larger intensity than the type 1 infection. If, instead, the initial type 1 infection is restricted to the negative x₁-axis, it is shown that the type 2 infection at the origin can also grow unboundedly when the infection types have the same intensity.

In many applications, it is important to be able to sample paths of SDEs conditional on observations of various kinds. This paper studies SPDEs which solve such sampling problems. The SPDE may be viewed as an infinite-dimensional analogue of the Langevin equation used in finite-dimensional sampling. In this paper, conditioned nonlinear SDEs, leading to nonlinear SPDEs for the sampling, are studied. In addition, a class of preconditioned SPDEs is studied, found by applying a Green’s operator to the SPDE in such a way that the invariant measure remains unchanged; such infinite dimensional evolution equations are important for the development of practical algorithms for sampling infinite dimensional problems.

The resulting SPDEs provide several significant challenges in the theory of SPDEs. The two primary ones are the presence of nonlinear boundary conditions, involving first order derivatives, and a loss of the smoothing property in the case of the pre-conditioned SPDEs. These challenges are overcome and a theory of existence, uniqueness and ergodicity is developed in sufficient generality to subsume the sampling problems of interest to us. The Gaussian theory developed in Part I of this paper considers Gaussian SDEs, leading to linear Gaussian SPDEs for sampling. This Gaussian theory is used as the basis for deriving nonlinear SPDEs which affect the desired sampling in the nonlinear case, via a change of measure.

We consider the random walk on a simple point process on ℝ^d, d≥2, whose jump rates decay exponentially in the α-power of jump length. The case α=1 corresponds to the phonon-induced variable-range hopping in disordered solids in the regime of strong Anderson localization. Under mild assumptions on the point process, we show, for α∈(0, d), that the random walk confined to a cubic box of side L has a.s. Cheeger constant of order at least L⁻¹ and mixing time of order L². For the Poisson point process, we prove that at α=d, there is a transition from diffusive to subdiffusive behavior of the mixing time.

Given a closed, bounded convex set $\mathcal{W}\subset{\mathbb {R}}^{d}$ with nonempty interior, we consider a control problem in which the state process W and the control process U satisfy $$W_{t}=w_{0}+\int_{0}^{t}\vartheta(W_{s})\,ds+\int_{0}^{t}\sigma(W_{s})\,dZ_{s}+GU_{t}\in \mathcal{W},\qquad t\ge0,$$ where Z is a standard, multi-dimensional Brownian motion, $\vartheta,\sigma\in C^{0,1}(\mathcal{W})$, G is a fixed matrix, and $w_{0}\in\mathcal{W}$. The process U is locally of bounded variation and has increments in a given closed convex cone $\mathcal{U}\subset{\mathbb{R}}^{p}$. Given $g\in C(\mathcal{W})$, κ∈ℝ^p, and α>0, consider the objective that is to minimize the cost $$J(w_{0},U)\doteq\mathbb{E}\biggl[\int_{0}^{\infty}e^{-\alpha s}g(W_{s})\,ds+\int_{[0,\infty)}e^{-\alpha s}\,d(\kappa\cdot U_{s})\biggr]$$ over the admissible controls U. Both g and κ⋅u ($u\in\mathcal{U}$) may take positive and negative values. This paper studies the corresponding dynamic programming equation (DPE), a second-order degenerate elliptic partial differential equation of HJB-type with a state constraint boundary condition. Under the controllability condition $G\mathcal{U}={\mathbb{R}}^{d}$ and the finiteness of $\mathcal{H}(q)=\sup_{u\in\mathcal{U}_{1}}\{-Gu\cdot q-\kappa\cdot u\}$, q∈ℝ^d, where $\mathcal{U}_{1}=\{u\in\mathcal{U}:|Gu|=1\}$, we show that the cost, that involves an improper integral, is well defined. We establish the following: (i) the value function for the control problem satisfies the DPE (in the viscosity sense), and (ii) the condition $\inf_{q\in{\mathbb{R}}^{d}}\mathcal{H}(q)< 0$ is necessary and sufficient for uniqueness of solutions to the DPE. The existence and uniqueness of solutions are shown to be connected to an intuitive “no arbitrage” condition.

Our results apply to Brownian control problems that represent formal diffusion approximations to control problems associated with stochastic processing networks.

We study a discrete time spatial branching system on ℤ^d with logistic-type local regulation at each deme depending on a weighted average of the population in neighboring demes. We show that the system survives for all time with positive probability if the competition term is small enough. For a restricted set of parameter values, we also obtain uniqueness of the nontrivial equilibrium and complete convergence, as well as long-term coexistence in a related two-type model. Along the way we classify the equilibria and their domain of attraction for the corresponding deterministic coupled map lattice on ℤ^d.