NBC News Scripts
WBAP-TV (Television station : Fort Worth, Tex.)
1954-12-31
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Brownian motion Brownian convex hull
We consider a countable system of stochastic differential equation. Euler scheme for approximating these solutions is used, and the global error is estimated. Solutions are approximated by means of a process which takes values in a finite dimensional space. Finally, we expand the global error for a class of smooth functions in powers of the discretization step size.
Research supported in part by the NSF Grant DMS 8702620.
Burdzy's research supported in part by NSF grant DMS 91-00244, Fondef grant F-11 and AMS Centennial
Research Fellowship. San Martin's research supported in part by Fondecyt grant 1940330.
We introduce two stochastic chemostat models consisting in a coupled population-nutrient process reflecting the interaction between the nutrient and the bacterias in the chemostat with finite volume. The nutrient concentration evolves continuously but depending on the population size, while the population size is a birth and death process with coefficients depending on time through the nutrient concentration. The nutrient is shared by the bacteria and creates a regulation of the bacterial population size. The latter and the fluctuations due to the random births and deaths of individuals make the population go almost surely to extinction. Therefore, we are interested in the long time behavior of the bacterial population conditioned to the non-extinction. We prove the global existence of the process and its almost sure extinction. The ex...
In this paper we study one-dimensional BSDE's whose coefficient f is monotonic in y and non-Lipschitz in z. We obtain a general existence result when f has at most quadratic growth in z and is bounded. We study the special case f (t, y, z) = vertical bar z vertical bar(p) where p is an element of (1, 2]. Finally, we study the case f has a linear growth in z, general growth in y and is not necessarily bounded.
We study the probabilistic evolution of a birth and death continuous time measure-valued process with mutations and ecological interactions. The individuals are characterized by (phenotypic) traits that take values in a compact metric space. Each individual can die or generate a new individual. The birth and death rates may depend on the environment through the action of the whole population. The offspring can have the same trait or can mutate to a randomly distributed trait. We assume that the population will be extinct almost surely. Our goal is the study, in this infinite dimensional framework, of quasi-stationary distributions when the process is conditioned on non-extinction. We firstly show in this general setting, the existence of quasi-stationary distributions. This result is based on an abstract theorem proving the existence o...
In this paper, we study quasi-stationarity for a large class of Kolmogorov diffusions. The main novelty here is that we allow the drift to go to $- \infty$ at the origin, and the diffusion to have an entrance boundary at $+\infty$. These diffusions arise as images, by a deterministic map, of generalized Feller diffusions, which themselves are obtained as limits of rescaled birth--death processes. Generalized Feller diffusions take nonnegative values and are absorbed at zero in finite time with probability $1$. An important example is the logistic Feller diffusion. We give sufficient conditions on the drift near $0$ and near $+ \infty$ for the existence of quasi-stationary distributions, as well as rate of convergence in the Yaglom limit and existence of the $Q$-process. We also show that under these conditions, there is exactly one qua...
Both the physiological and the pathological morphogenetic processes that we can meet in embryogenesis, neogenesis and degenerative dysgenesis present common features: they are ruled by three different kinds of mechanisms, one related to cell migration, the second to cell differentiation and the third to cell proliferation. We deal here with an application to the cambial growth which essentially involves the third type of mechanism. Woody plants produce secondary tissue (secondary xylem and phloem) from a meristematic tissue called vascular cambium, responsible for the radial growth of a tree. This paper focuses on the formation of secondary xylem, considered in two dimensions in a cross-section framework. A new discrete modelling approach is used, based on the cellular scale, in order to attain a more accurate understanding of how the ...
