Interacting locally regulated diffusions [Elektronische Ressource] / von Martin Hutzenthaler

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• Mathematics

Informations

• Publié par: goethe_universitat_frankfurt_am_main
• Publié le: 01 janvier 2007
• Nombre de lectures: 9
• Langue: English

Extrait

Interacting locally
regulated di usions
Dissertation
zur Erlangung des Doktorgrades
der Naturwissenschaften
vorgelegt beim Fachbereich Informatik und Mathematik
der Johann Wolfgang Goethe-Universit at
in Frankfurt am Main
von
Martin Hutzenthaler
aus Landshut
Frankfurt am Main 2007
(D 30)ii
vom Fachbereich Informatik und Mathematik
der Johann Wolfgang Goethe { Universit at als Dissertation angenommen.
Dekan: Prof. Dr.-Ing. Detlef Kr omker
Gutachter: Prof. Dr. Anton Wakolbinger
Prof. Dr. G otz Kersting
Prof. Dr. Alison Etheridge
Datum der Disputation:A mathematician is a device
for turning co ee into theorems.
Paul Erd} os (1913-1996)iv
Acknowledgments
First and foremost, I wish to express my deepest gratitude to my advisor
Prof. Dr. Anton Wakolbinger for his excellent mentoring. In the last three years,
I enjoyed to bene t from his intuition and brilliance. In uncounted meetings,
he inspired and stimulated me with ideas and pictures for new approaches. He
considerably improved my way of writing papers by countless detailed remarks.
Furthermore, especially in frustrating moments, Anton Wakolbinger encouraged
me with his positive attitude to go on.
Next, I thank my co-authors. The main part of Chapter 2 is joint work with
Anton Wakolbinger ([17]). The main part of Chapter 4 is joint work with Roland
Alkemper ([1]).
Furthermore, I am grateful to Achim Klenke and three anonymous referees
for detailed remarks and useful suggestions which have considerably improved the
quality of the above mentioned papers.
This thesis pro ted much by valuable discussions with Don Dawson, Frank den
Hollander and Jan Swart. Their brightness and their skills produced useful ideas.
In addition, Chapter 2 is inspired by ideas from the paper [2] of Siva Athreya and
Jan Swart.
Last but not least, I thank my next-door colleague Wolfgang Angerer, who
moved to Mexico in 2006, for his patience to listen to all my questions and for all
the discussions with him.Contents
1 Introduction and main results 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.3 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2 Local extinction and ergodic behaviour 23
2.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.2 The upper invariant measure. Proof of Theorem 1 . . . . . . . . . . 30
2.3 The mean eld model . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.4 Comparison with the mean eld model. Proof of Theorem 2 . . . . 36
2.5 Self-duality. Proof of Theorem 3 . . . . . . . . . . . . . . . . . . . . 40
2.6 Convergence to the upper invariant measure. Proof of Theorem 5 . 42
3 The Virgin Island Model 45
3.1 Excursions from a trap of one-dimensional di usions. Proof of The-
orem 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.2 Proof of Theorem 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.3 Recursion for the Virgin Island process . . . . . . . . . . . . . . . . 60
3.4 Extinction and survival in the Virgin Island Model. Proof of The-
orem 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4 Graphical representation of two duality relations 67
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.2 Dual basic mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . 70
4.3 Prototype duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
4.4 Various scalings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
4.5 Weak convergence of processes . . . . . . . . . . . . . . . . . . . . . 80
Bibliography 83
Deutsche Zusammenfassung 87
vvi CONTENTSN
Chapter 1
Introduction and main results
1.1 Introduction
In naturally reproducing populations one usually encounters an average number
of more than one o spring per individual. However, given non-extinction, classi-
cal supercritical branching processes grow beyond all bounds. This is unrealistic
because of bounded resources.
An e cient counteraction to unbounded population growth is achieved by a
population-size dependent regulation of the reproduction dynamics. An example
is the so called logistic branching process (Lambert [23]) in which, in addition to
the \natural" births and deaths in a supercritical branching mechanism, there are
deaths resulting from a competition between any two individuals in the population.
In Feller’s di usion limit, this leads to a negative drift term which is proportional
to the squared population size. To be more precise, for N 1 and b;d;; > 0,
Nlet (Z ) be a pure birth-death process with state space where each particlet0 0t
b dsplits into two particles at rate + , each particle dies at rate + and each
N N

ordered pair of particles coalesces into one at rate . All these events2N
NZ0occur independently of each other. If converges weakly to Z asN!1 then0N
N Z
tN converges weakly to Z as N!1 where (Z ) is the solution oft t t0N t0 t0
p
2(1.1) dZ = (b d)Z dt Z dt + 2