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Models of populations in which a type or location, represented by a point in a metric space E, is associated with each individual in the population are considered. A population process is neutral if the chances of an individual replicating or dying do not depend on its type. Measurevalued processes are obtained as infinite population limits for a large… (More)

- Jin Feng, Thomas G. Kurtz
- 2000

The theory of large deviations deals with the estimation of small probabilities, particularly those that are exponentially small in some natural parameter. The general goal is to identify the constant in the exponent that dictates the exponential rate of decay. In many situations the constant can be “explicitly” calculated and turns out often to be… (More)

- David F. Anderson, Gheorghe Craciun, Thomas G. Kurtz
- Bulletin of mathematical biology
- 2010

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… (More)

Let W denote a standard Wiener process with W0 = 0. For a variety of reasons, it is desirable to have a notion of an integral ∫ 1 0 HsdWs, where H is a stochastic process; or more generally an indefinite integral ∫ t 0 HsdWs, 0 ≤ t < ∞. If H is a process with continuous paths, an obvious way to define a stochastic integral is by a limit of sums: let πn[0,… (More)

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… (More)

We perform an error analysis for numerical approximation methods of continuous time Markov chain models commonly found in the chemistry and biochemistry literature. The motivation for the analysis is to be able to compare the accuracy of different approximation methods and, specifically, Euler tau-leaping and midpoint tau-leaping. We perform our analysis… (More)

- Thomas G. Kurtz
- 1998

Let X be a Markov process with generator A and let Y (t) = γ(X(t)). The conditional distribution πt of X(t) given σ(Y (s) : s ≤ t) is characterized as a solution of a filtered martingale problem. As a consequence, we obtain a generator/martingale problem version of a result of Rogers and Pitman on Markov functions. Applications include uniqueness of… (More)

- Thomas G. Kurtz
- 2000

General models for workload input into service systems are considered. Scaling limit theorems appropriate for the formulation of fluid and heavy traffic approximations for systems driven by these inputs are given. Under appropriate assumptions, it is shown that fractional Brownian motion can be obtained as the limiting workload input process. Motivation for… (More)

We give complete proofs of the theorem of convergence of types and the Kesten-Stigum theorem for multi-type branching processes. Very little analysis is used beyond the strong law of large numbers and some basic measure theory. Consider a multi-type Galton-Watson branching process with J types. Let L be a random variable representing the number of particles… (More)

A reaction network is a chemical system involving multiple reactions and chemical species. Stochastic models of such networks treat the system as a continuous time Markov chain on the number of molecules of each species with reactions as possible transitions of the chain. In many cases of biological interest some of the chemical species in the network are… (More)