Law of Large Numbers for a Class of Superdiffusions

Under spectral conditions, we prove a LLN type result for superdiffusions, where the convergence is meant in probability. The main tool is a space-time H-transformation

Turning a coin over instead of tossing it

Given a sequence of numbers $\{p_n\}$ in $[0,1]$, consider the following experiment. First, we flip a fair coin and then, at step $n$, we turn the coin over to the other side with probability $p_n$, $n\ge 2$. What can we say about the distribution of the empirical frequency of heads as $n\to\infty$? We show that a number of phase transitions take place as the turning gets slower (i.e. $p_n$ is getting smaller), leading first to the breakdown of the Central Limit Theorem and then to that of the Law of Large Numbers. It turns out that the critical regime is $p_n=\text{const}/n$. Among the scaling limits, we obtain Uniform, Gaussian, Semicircle and Arcsine laws

Survival asymptotics for branching random walks in IID environments

We first study a model, introduced recently in \cite{ES}, of a critical branching random walk in an IID random environment on the $d$-dimensional integer lattice. The walker performs critical (0-2) branching at a lattice point if and only if there is no `obstacle' placed there. The obstacles appear at each site with probability $p\in [0,1)$ independently of each other. We also consider a similar model, where the offspring distribution is subcritical. Let $S_n$ be the event of survival up to time $n$. We show that on a set of full $\mathbb P_p$-measure, as $n\to\infty$, (i) Critical case: P^{\omega}(S_n)\sim\frac{2}{qn}; (ii) Subcritical case: P^{\omega}(S_n)= \exp\left[\left( -C_{d,q}\cdot \frac{n}{(\log n)^{2/d}} \right)(1+o(1))\right], where $C_{d,q}>0$ does not depend on the branching law. Hence, the model exhibits `self-averaging' in the critical case but not in the subcritical one. I.e., in (i) the asymptotic tail behavior is the same as in a "toy model" where space is removed, while in (ii) the spatial survival probability is larger than in the corresponding toy model, suggesting spatial strategies. We utilize a spine decomposition of the branching process as well as some known results on random walks.Comment: 2 figure

Technology Selection and Appropriate Technology

This paper provides a formal model of technology choice by a single region. Case studies have indicated that the technology acquired by LDCs often seem unsuitable, although the criteria for suitability are often unclear. The reasons which are presented for inappropriateness of the selection often rely more on political arguments then economic ones, or treat the recipient country as a passive actor in the whole process. Can a technology actively selected by a recipient country ever by inappropriate, assuming factor cost ratios represent true relative values? A model presented by Evenson and Bingswanger (1978) indicates that a technology developed in one economic or physical environment may be 'appropriate' to a second, very different environment if the second environment can generate a very limited range of technological possibilities on its own. Ranis (1978) has emphasized the importance of information on technological alternatives flowing smoothly and accurately within the system and the need to acquire capacity for adaptive research. Both these approaches recognize the importance of indigenous research capacity, although Ranis accords more emphasis to friction and proper incentives within the system. Barring policy and management problems, their conclusions appear to be that technology choice will be efficient--the appearance of inappropriateness stems from the lack of explicit recognition of the constraints on technology generation in the system. The model presented below builds on the early models of rational technology selection of Evenson- Binswanger and Ranis. It shares common elements with the Evenson-Binswanger model and may be regarded as a generalization of their model. It goes further, however, in several crucial aspects. It allows the extent of both adaptive and independent research to be choice variables in the technology acquisition decision. It allows for selection out of a continuum of technologies which differ in the environments for which they were designed. It allows for limits to the extent to which technologies can be adapted across environments and allows for losses because of incomplete adaption. The public goodnature of research plays a critical role in determining the efficiency of resource allocation as well. The model presented immediately below is couched in terms relating to agricultural technology. A reason for first presenting a model of agricultural technology selection is that many of the conceptual issues possess more intuitive natural interpretations. A second section will consider the impact of market structure on the development of technology, and a third section will broaden the basic model of agricultural technology development to one which encompasses certain types of fixed capital investment. A fourth section discusses testing of the model.Research and Development/Tech Change/Emerging Technologies,

The compact support property for measure-valued processes

The purpose of this article is to give a rather thorough understanding of the compact support property for measure-valued processes corresponding to semi-linear equations of the form $$\begin{aligned}& u_t=Lu+\beta u-\alpha u^p \text{in} R^d\times (0,\infty), p\in(1,2]; &u(x,0)=f(x) \text{in} R^d; &u(x,t)\ge0 \text{in} R^d\times[0,\infty). \end{aligned}$$ In particular, we shall investigate how the interplay between the underlying motion (the diffusion process corresponding to $L$) and the branching affects the compact support property. In \cite{EP99}, the compact support property was shown to be equivalent to a certain analytic criterion concerning uniqueness of the Cauchy problem for the semi-linear parabolic equation related to the measured valued process. In a subsequent paper \cite{EP03}, this analytic property was investigated purely from the point of view of partial differential equations. Some of the results obtained in this latter paper yield interesting results concerning the compact support property. In this paper, the results from \cite{EP03} that are relevant to the compact support property are presented, sometimes with extensions. These results are interwoven with new results and some informal heuristics. Taken together, they yield a rather comprehensive picture of the compact support property. \it Inter alia\rm, we show that the concept of a measure-valued process \it hitting\rm a point can be investigated via the compact support property, and suggest an alternate proof of a result concerning the hitting of points by super-Brownian motion