Control problems in stochastic queuing networks are hard to solve. However, it is well known that the fluid model provides a useful approximation to the stochastic network.We will formulate a general class of control problems in stochastic queuing networks and consider the corresponding fluid optimization problem ($F$) which is a deterministic control problem and often easy to solve. Contrary to previous literature, our cost rate function is rather general.The value function of ($F$) provides an asymptotic lower bound on the value function of the stochastic network under fluid scaling. Moreover, we can construct from the optimal control of ($F$) a so-called tracking policy for the stochastic queuing network which achieves the lower bound as the fluid scaling parameter tends to $\infty$. In this case we say that the tracking policy is asymptotically optimal. This statement is true for multiclass queuing networks and admission and routing problems.The convergence is monotone under some convexity assumptions. The tracking policy approach also shows that a given fluid model solution can be attained as a fluid limit of the original discrete model.

Consider a queue with a stochastic fluid input process modeled as fractional Brownian motion (fBM).When the queue is stable, we prove that the maximum of the workload process observed over an interval of length $t$ grows like $\gamma(\log t)^{1/(2-2H)}, where $H > ½$ is the self-similarity index (also known as the Hurst parameter) that characterizes the fBM and can be explicitly computed. Consequently, we also have that the typical time required to reach a level $b$ grows like $\exp{b^{2(1-H)}}.We also discuss the implication of these results for statistical estimation of the tail probabilities associated with the steady-state workload distribution.

W show that the motion of a particle advected by a random Gaussian velocity field with long-range correlations converges to a fractional Brownian motion in the long time limit.

In this paper we investigate the structure of the equilibriumstate of three-dimensional catalytic super-Brownian motion where the catalyst is itself a classical super-Brownian motion.We show that the reactant has an infinite local biodiversity or genetic abundance. This contrasts to the finite local biodiversity of the equilibriumof classical super-Brownian motion. Another question we address is that of extinction of the reactant in finite time or in the long-time limit in dimensions $d = 2,3$. Here we assume that the catalyst starts in the Lebesgue measure and the reactant starts in a finite measure.We show that there is extinction in the long-time limit if $d = 2 or 3$. There is, however, no finite time extinction if $d = 3$ (for $d = 2$, this problem is left open).This complements a result of Dawson and Fleischmann for $d = 1$ and again contrasts the behaviour of classical super-Brownian motion. As a key tool for both problems, we show that in $d = 3$ the reactant matter propagates everywhere in space immediately.

Let $X_1, X_2,\dots, X_n$ be a sequence of integer-valued random variables that are either associated or negatively associated.We present a simple upper bound for the distance between the distribution of the sumof $X_i$ and a sum of $n$ independent randomvariables with the same marginals as $X_i$. An upper bound useful for establishing a compound Poisson approximation for $\Sigma_{i=1}^nX_i$ is also provided. The new bounds are of the same order as the much acclaimed Stein–Chen bound.

With a given transformation on a finite domain, we associate a three-dimensional distribution function describing the component size, cycle length and trajectory length of each point in the domain.We then consider a random transformation on the domain, in which images of points are independent and identically distributed. The three-dimensional distribution function associated with this random transformation is itself random. We show that, under a simple homogeneity condition on the distribution of images, and with a suitable scaling, this random distribution function has a limit law as the number of points in the domain tends to $\infty$. The proof is based on a Poisson approximation technique for matches in an urn model. The result helps to explain the behavior of computer implementations of chaotic dynamical systems.

By using mixed percolation as a bridge between site and bond percolation, we derive a new inequality between the critical points of these processes that is optimal in a certain sense.We also extend a result on the crossover exponent of bond-diluted Potts models to site-diluted Potts models. Some newresults about the critical line in mixed percolation are also proved.

We consider the Navier–Stokes equation in dimension 2 and more precisely the vortex equation satisfied by the curl of the velocity field.We show the relation between this equation and a nonlinear stochastic differential equation. Next we use this probabilistic interpretation to construct approximating interacting particle systems which satisfy a propagation of chaos property: the laws of the empirical measures tend, as the number of particles tends to $\infty$, to a deterministic law for which marginals are solutions of the vortex equation.This pathwise result justifies completely the vortex method introduced by Chorin to simulate the solutions of the vortex equation.Our approach is inspired by Marchioro and Pulvirenti and we improve their results in a pathwise sense.

We describe the averaged over the disordered dynamics for the random field Curie–Weiss model. We consider both the magnetization and the full spin dynamics.Our approach is based on spectral asymptotics and includes results on the random fluctuations of eigenvalues and eigenvectors.

In this paper we construct a kernel estimator of a time-varying coefficient of a strongly elliptic partial differential operator in a stochastic parabolic equation.The equation is assumed diagonalizable; that is, all the operators have a common system of eigenfunctions.The mean-square convergence of the estimator is established. The rate of convergence is determined both by the smoothness of the true coefficient and by the asymptotics of the eigenvalues of the operators in the equation.

This is the fourth in a series of papers devoted to the study of the large deviations of a Wright –Fisher process modeling the genetic evolution of a reproducing population.Variational considerations imply that if the process undergoes a large deviation, then it necessarily follows closely a definite path from its original to its current state. The favored paths were determined previously for a one-dimensional process subject to one-way mutation or natural selection, respectively, acting on a faster time scale than random genetic drift. The present paper deals with a general $d$-dimensional Wright–Fisher process in which any mutation or selection forces act on a time scale no faster than that of genetic drift. If the states of the process are represented as points on a $d$-sphere, then it can be shown that the position of a subcritically scaled process at a fixed “time” $T$ satisfies a large-deviation principle with rate function proportional to the square of the length of the great circle arc joining this position with the initial one (Hellinger–Bhattacharya distance). If a large deviation does occur, then the process follows with near certainty this arc at constant speed. The main technical problem circumvented is the degeneracy of the covariance matrix of the process at the boundary of the state space.

While the extreme value statistics for i.i.d data is well developed, much less is known about the asymptotic behavior of statistical procedures in the presence of dependence.We establish convergence of tail empirical processes to Gaussianlimits for $\beta$-mixing stationary time series. As a consequence, one obtains weighted approximations of the tail empirical quantile function that is based on a random sequence with marginal distribution belonging to the domain of attraction of an extreme value distribution. Moreover, the asymptotic normality is concluded for a large class of estimators of the extreme value index. These results are applied to stationary solutions of a general stochastic difference equation.

The p-shuffle is a natural generalization of the dovetail shuffle. It is defined as follows. First, the deck is cut into a top stack and a bottom stack so that the distribution of the size of the top stack is Binomial $(N, p)$, where $N$ is the total number of cards in the deck.Then, conditional on the outcome of the cut,the two stacks are “riffled” in such a way that all possible riffles (interleavings) of these two stacks are equally likely. The main result of the paper is an asymptotic $(N \to \infty)$ bound on the number of repetitions needed to “randomize” the deck.