Thomas G. Kurtz

The Relationship between Stochastic and Deterministic Models for Chemical Reactions

The Markov chain and ordinary differential equation models for chemical reaction systems are compared. It is shown that if the volume of the reaction system is taken into account in an appropriate way in the formulation of the Markov chain model, then the o.d.e. model is the infinite volume limit of the Markov chain model. A central limit theorem is also given for the deviation of the Markov chain model from the o.d.e. model.

Strong approximation theorems for density dependent Markov chains

A variety of continuous parameter Markov chains arising in applied probability (e.g. epidemic and chemical reaction models) can be obtained as solutions of equations of the form where l[set membership, variant]Zt, the Y1 are independent Poisson processes, and N is a parameter with a natural interpretation (e.g. total population size or volume of a reacting solution). The corresponding deterministic model, satisfies X(t)=x0+ [integral operator]t0 [summation operator] lf1(X(s))ds Under very general conditions limN-->[infinity]XN(t)=X(t) a.s. The process XN(t) is compared to the diffusion processes given by and Under conditions satisfied by most of the applied probability models, it is shown that XN,ZN and V can be constructed on the same sample space in such a way that and

The changing nature of network traffic: scaling phenomena

In this paper, we report on some preliminary results from an in-depth, wavelet-based analysis of a set of high-quality, packet-level traffic measurements, collected over the last 6-7 years from a number of different wide-area networks (WANs). We first validate and confirm an earlier finding, originally due to Paxson and Floyd [14], that actual WAN traffic is consistent with statistical self-similarity for sufficiently large time scales. We then relate this large-time scaling phenomenon to the empirically observed characteristics of WAN traffic at the level of individual connections or applications. In particular, we present here original results about a detailed statistical analysis of Web-session characteristics, and report on an intriguing scaling property of measured WAN traffic at the transport layer (i.e., number of TCP connection arrivals per time unit). This scaling property of WAN traffic at the TCP layer was absent in the pre-Web period but has become ubiquitous in todays WWW-dominated WANs and is a direct consequence of the ever-increasing popularity of the Web (WWW) and its emergence as the major contributor to WAN traffic. Moreover, we show that this changing nature of WAN traffic can be naturally accounted for by self-similar traffic models, primarily because of their ability to provide physical explanations for empirically observed traffic phenomena in a networking context. Finally, we provide empirical evidence that actual WAN traffic traces also exhibit scaling properties over small time scales, but that the small-time scaling phenomenon is distinctly different from the observed large-time scaling property. We relate this newly observed characteristic of WAN traffic to the effects that the dominant network protocols (e.g., TCP) and controls have on the flow of packets across the network and discuss the potential that multifractals have in this context for providing a structural modeling approach for WAN traffic and for capturing in a compact and parsimonious manner the observed scaling phenomena at large as well as small time scales.

A limit theorem for perturbed operator semigroups with applications to random evolutions

Let U(t) and S(t) be strongly continuous contraction semigroups on a Banach space L with infinitesimal operators A and B, respectively. Suppose the closure of A + αB generates a semigroup Tα(t). The behavior of Tα(t) as α goes to infinity is examined. In particular, suppose S(t) converges strongly to P. If the closure of PA generates a semigroup T(t) on R(P), then Tα(t) goes to T(t) on R(P). If PA = 0 and if BVf = −f for feN(P), conditions are given that imply Tα(αt) converges on R(P) to a semigroup generated by the closure of PAVA. The results are used to obtain new and known limit theorems for random evolutions, which in turn give approximation theorems for diffusion processes.

Fleming-Viot processes in population genetics

Fleming and Viot [Indiana Univ. Math. J., 28 (1979), pp. 817–843] introduced a class of probability-measure-valued diffusion processes that has attracted the interest of both pure and applied probabilists. This paper surveys the subject of Fleming–Viot processes as it relates to population genetics. Topics include:1. Introduction.2. Some measure-valued Markov chains.2.1. A diploid model. 2.2. The Wright–Fisher model.2.3. A Moran model.3. The Fleming–Viot process: characterization.4. Convergence.5. Ergodicity.6. An infinite particle system.7. Bounded mutation operators.8. Reversibility.9. Examples.9.1. Continuous-state stepwise-mutation model.9.2. Infinitely-many-neutral-alleles model.9.3. Infinitely-many-neutral-alleles model with ages.9.4. Two-locus model with recombination.9.5. n-locus model with gene conversion.9.6. Infinitely-many-sites model without recombination.9.7. Infinitely-many-neutral-alleles model with allelic genealogies.

Continuous Time Markov Chain Models for Chemical Reaction Networks

A reaction network is a chemical system involving multiple reactions and chemical species. The simplest stochastic models of such networks treat the system as a continuous time Markov chain with the state being the number of molecules of each species and with reactions modeled as possible transitions of the chain. This chapter is devoted to the mathematical study of such stochastic models. We begin by developing much of the mathematical machinery we need to describe the stochastic models we are most interested in. We show how one can represent counting processes of the type we need in terms of Poisson processes. This random time-change representation gives a stochastic equation for continuous-time Markov chain models. We include a discussion on the relationship between this stochastic equation and the corresponding martingale problem and Kolmogorov forward (master) equation. Next, we exploit the representation of the stochastic equation for chemical reaction networks and, under what we will refer to as the classical scaling, show how to derive the deterministic law of mass action from the Markov chain model. We also review the diffusion, or Langevin, approximation, include a discussion of first order reaction networks, and present a large class of networks, those that are weakly reversible and have a deficiency of zero, that induce product-form stationary distributions. Finally, we discuss models in which the numbers of molecules and/or the reaction rate constants of the system vary over several orders of magnitude. We show that one consequence of this wide variation in scales is that different subsystems may evolve on different time scales and this time-scale variation can be exploited to identify reduced models that capture the behavior of parts of the system. We will discuss systematic ways of identifying the different time scales and deriving the reduced models.

Existence of Markov Controls and Characterization of Optimal Markov Controls

Given a solution of a controlled martingale problem it is shown under general conditions that there exists a solution having Markov controls which has the same cost as the original solution. This result is then used to show that the original stochastic control problem is equivalent to a linear program over a space of measures under a variety of optimality criteria. Existence and characterization of optimal Markov controls then follows. An extension of Echeverrias theorem characterizing stationary distributions for (uncontrolled) Markov processes is obtained as a corollary. In particular, this extension covers diffusion processes with discontinuous drift and diffusion coefficients.

Particle representations for a class of nonlinear SPDEs

An infinite system of stochastic differential equations for the locations and weights of a collection of particles is considered. The particles interact through their weighted empirical measure, V, and V is shown to be the unique solution of a nonlinear stochastic partial differential equation (SPDE). Conditions are given under which the weighted empirical measure has an L2-density with respect to Lebesgue measure.

Weak convergence and local stability properties of fixed step size recursive algorithms

A recursive equation that subsumes several common adaptive filtering algorithms is analyzed for general stochastic inputs and disturbances by relating the motion of the parameter estimate errors to the behavior of an unforced deterministic ordinary differential equation (ODE). The ODEs describing the motion of several common adaptive filters are examined in some simple settings, including the least mean square (LMS) algorithm and all three of its signed variants (the signed regressor, the signed error, and the sign-sign algorithms). Stability and instability results are presented in terms of the eigenvalues of a correlation-like matrix. This generalizes known results for LMS, signed regressor LMS, and signed error LMS, and gives new stability criteria for the sign-sign algorithm. The ability of the algorithms to track moving parameterizations can be analyzed in a similar manner, by relating the time varying system to a forced ODE. The asymptotic distribution about the forced ODE is an Ornstein-Uhlenbeck process, the properties of which can be described in a straightforward manner. >

Product-form stationary distributions for deficiency zero chemical reaction networks.

We consider stochastically modeled chemical reaction systems with mass-action kinetics and prove that a product-form stationary distribution exists for each closed, irreducible subset of the state space if an analogous deterministically modeled system with mass-action kinetics admits a complex balanced equilibrium. Feinberg’s deficiency zero theorem then implies that such a distribution exists so long as the corresponding chemical network is weakly reversible and has a deficiency of zero. The main parameter of the stationary distribution for the stochastically modeled system is a complex balanced equilibrium value for the corresponding deterministically modeled system. We also generalize our main result to some non-mass-action kinetics.