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Analysis of a class of branching particle systems with spatial pairwise interactions
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Matthews, Paul (2002) Analysis of a class of branching particle systems with spatial pairwise interactions. PhD thesis, University of Warwick.
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Official URL: http://webcat.warwick.ac.uk/record=b3781108
Abstract
The main aim of this work is to study a particular type of one-dimensional interacting particle system. The behaviour of the particles in the process is governed by three mechanisms: diffusive movement, independent single-particle branching and two-particle spatial interaction. It is this third feature, the pairwise interaction, which makes the model new and challenging, and gives rise to some exciting problems. It will be explained later that in most cases the process will not be attractive and hence many of the tools usually used in the analysis of interacting systems will not be applicable.
It is natural to begin by asking how such processes can be constructed, and whether they can in fact be constructed at all. As the number of pairwise interactions can potentially scale as the square of the population, the possibility of explosion in finite time must be considered. The initial work deals with finite systems of particles, but later models consist of infinitely many particles distributed on the real line; in this case it is possible that the model cannot be defined for any time except t = 0.
The majority of the following work concentrates on those cases in which the branching forms a ‘growth’ mechanism whilst the two-particle interactions are ‘reductive’. This means that the expected number of offspring from a branching event is greater than one, whilst the expected number of offspring from a pairwise interaction is less than two. These models allow a great deal of intuitive understanding, which is then consolidated through the mathematics.
Once a construction of the process has been provided, attention naturally turns to the existence and uniqueness of associated stationary distributions. In the case of pure branching diffusions, with no interactions between particles, such questions have been fully answered. In these interacting models such questions can be far more difficult.
The motivation for the study of such processes is two-fold. The problems are interesting and beautiful in their own right. Very few examples of non-attractive interacting systems have been successfully analysed in anything other than a numerical way. Complementing this is the fact that such models appear as dual processes to certain white-noise driven stochastic partial differential equations (SPDEs). This duality will be discussed in detail in this paper and it will be shown that information about the particle system yields information about the corresponding SPDE, and vice-versa.
Item Type: | Thesis (PhD) | ||||
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Subjects: | Q Science > QA Mathematics | ||||
Library of Congress Subject Headings (LCSH): | Branching processes, Stochastic differential equations, Particles (Nuclear physics) -- Mathematical models | ||||
Official Date: | 12 February 2002 | ||||
Dates: |
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Institution: | University of Warwick | ||||
Theses Department: | Mathematics Institute | ||||
Thesis Type: | PhD | ||||
Publication Status: | Unpublished | ||||
Supervisor(s)/Advisor: | Tribe, Roger | ||||
Format of File: | |||||
Extent: | 203 leaves | ||||
Language: | eng |
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