We study a spatial branching model, where the underlying motion is d-dimensional (d≥1) Brownian motion and the branching rate is affected by a random collection of reproduction suppressing sets dubbed mild obstacles. The main result of this paper is the quenched law of large numbers for the population for all d≥1. We also show that the branching Brownian motion with mild obstacles spreads less quickly than ordinary branching Brownian motion by giving an upper estimate on its speed. When the underlying motion is an arbitrary diffusion process, we obtain a dichotomy for the quenched local growth that is independent of the Poissonian intensity. More general offspring distributions (beyond the dyadic one considered in the main theorems) as well as mild obstacle models for superprocesses are also discussed.

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## Scholarly Works (91 results)

In previous work of D. Turaev, A. Winter and the author, the Law of Large Numbers for the local mass of certain superdiffusions was proved under an ergodicity assumption. In this paper we go beyond ergodicity, that is we consider cases when the scaling for the expectation of the local mass is not purely exponential. Inter alia, we prove the analog of theWatanabe–Biggins LLN for super-Brownian motion.

Abstract

Pinsky [R.G. Pinsky, Transience, recurrence and local extinction properties of the support for supercritical finite measure-valued diffusions, Ann. Probab. 24 (1) 237–267] proved that the finite mass superdiffusion X corresponding to a semilinear operator exhibits local extinction if and only if l>0, where l is the generalized principal eigenvalue of the linear part of the operator. For the case when l>0, it has been shown in Engländer and Turaev [J. Engländer, D. Turaev, A scaling limit theorem for a class of superdiffusions, Ann. Probab. 30 (2) 683–722] that in law the superdiffusion locally behaves like exp[tl] times a non-negative non-degenerate random variable, provided that the linear part of the operator satisfies a certain spectral condition (‘product-criticality’), and that the intensity parameter and the starting measure are ‘not too large’.

In this article we will prove that the convergence in law used in the formulation in [J. Engländer, D. Turaev, A scaling limit theorem for a class of superdiffusions, Ann. Probab. 30 (2) 683–722] can actually be replaced by convergence in probability. Furthermore, instead of we will consider a general Euclidean domain .

As far as the proof of our main theorem is concerned, the heavy analytic method of [J. Engländer, D. Turaev, A scaling limit theorem for a class of superdiffusions, Ann. Probab. 30 (2) 683–722] is replaced by a different, simpler and more probabilistic one. We introduce a space–time weighted superprocess (H-transformed superprocess) and use it in the proof along with some elementary probabilistic arguments.

Let X be either the branching diffusion corresponding to the operator Lu + beta(u(2) - u) on D subset of or equal to R-d [where beta(x) greater than or equal to 0 and beta not equivalent to 0 is bounded from above] or the superprocess corresponding to the operator Lu + betau - alphau(2) on D subset of or equal to R-d (with alpha > 0 and beta is bounded from above but no restriction on its sign). Let lambda(c) denote the generalized principal eigenvalue for the operator L + beta on D. We prove the following dichotomy: either lambda(c) less than or equal to 0 and X exhibits local extinction or lambda(c) > 0 and there is exponential growth of mass on compacts of D with rate lambdac. For superdiffusions, this completes the local extinction criterion obtained by Pinsky [Ann. Probab. 24 (1996) 237-267] and a recent result on the local growth of mass under a spectral assumption given by Englander and Turaev [Ann. Probab. 30 (2002) 683-7221. The proofs in the above two papers are based on PDE techniques, however the proofs we offer are probabilistically conceptual. For the most part they are based on "spine" decompositions or "immortal particle representations" along with martingale convergence and the law of large numbers. Further they are generic in the sense that they work for both types of processes.

**Cities of Nature: Socio-natural Crisis and the Production of Space in New Orleans and Seattle**

**Nik Janos**

This dissertation shows how the seemingly *social* processes of urbanization are deeply entangled with seemingly *natural* processes, such as ecological, biological, and climatological ones. Using historical comparative methods and archival research, I compare urbanization in New Orleans and Seattle by looking at two apparently distinct social and ecological crises. One, in New Orleans I examine the entangled relationship between urbanization and hurricanes in Southeast Louisiana. Specifically, I trace the path dependent history between Hurricane Betsy in 1964 and Hurricane Katrina in 2005. Two, in Seattle I examine the entangled relationship between urbanization and the decimation of salmon populations in the Puget Sound. Specifically, I trace the path dependent history between a landmark 1974 court ruling called the Boldt Decision, which granted Native Americans half the salmon catch, and the 1999 Endangered Species Listing for Chinook salmon, which made the Seattle metropolitan area the first urban area to have an endangered species listing. Case one appears to be induced by nature, whereas case two by humans. But I say these are indistinct. The case studies indicate that the distinctions between what comes to us and what comes from us dissolves if you look at "natural disasters" as hybrid socio-natural processes. Furthermore, what people call disasters are not one-time events but rather crises long in the making. Overall, this dissertation tells a story about how the urbanization of New Orleans and Seattle has made the natural and social more entangled, which in turn has made some populations of humans and non-humans more vulnerable than others. The comparison of New Orleans and Seattle sheds light on how each urbanized region is shaped by particular social and ecological relations, but additionally, how city builders shape and are shaped by generalized strands of socio-natural entanglement: capitalist urbanization, deployment of technology, governance practices, and social participation. The comparison also illuminates how socio-natural relations and potential crises of the future are made by events and decision of the past as well as ones that unfold in the present.

Humans are social beings; people are predisposed to join groups, categorize the social world into groups, and prefer fellow in-group members over out-group members. Social groups in turn compete for individuals and especially for the resources of individuals to maintain the cultural practices and symbolic markers of the group. We modeled the effect of this competition on population level cooperation. Using game theoretic and network science methods, we found that groups would develop and maintain norms that restrict their members to join other groups. If every group can maintain such norms against every other group (the topology of the group-network is complete), the society is composed of closed communities which do not cooperate with each other. Changing the topology of the group-network can yield larger cooperating components within the population, because, in this case, members of antagonistic groups can join a third group, thereby allowing cooperation between them. The results suggest that the individuals’ ability to join more than one social group is crucial for maintaining cooperation in large populations.