Tree-valued Markov limit dynamics Habilitationsschrift Anita Winter Mathematisches Institut ...

228 Pages·2011·0.98 MB·English

Tree-valued Markov limit dynamics
Habilitationsschrift
Anita Winter
Mathematisches Institut
Universita(cid:127)t Erlangen{Nu(cid:127)rnberg
Bismarckstra(cid:25)e 11
2
91054 Erlangen
GERMANY
[email protected]

Contents
Many many thanks ... 5
Introduction 9
Part I: Real trees and metric measure spaces 10
Part II: Examples of prominent real trees and mm-spaces 13
Part III: Tree-valued Markov dynamics 15
Notes 20
Chapter 1. State spaces I: R-trees 21
1.1. The Gromov-strong topology 22
1.2. A complete metric: The Gromov-Hausdorff metric 22
1.3. Gromov-Hausdorff and the Gromov-strong topology coincide 24
1.4. Compact sets in X 25
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1.5. 0-hyperbolic spaces and R-trees 25
1.6. R-trees with 4 leaves 28
1.7. Length measure 29
1.8. Rooted R-trees 31
1.9. Rooted subtrees and trimming 34
1.10. Compact sets in T 40
1.11. Weighted R-trees 41
1.12. Distributions of random (weighted) real trees 47
Chapter 2. State spaces II: The space of metric measure trees 49
2.1. The Gromov-weak topology 50
2.2. A complete metric: The Gromov-Prohorov metric 53
2.3. Distance distribution and modulus of mass distribution 59
2.4. Compact sets in M 64
2.5. Gromov-Prohorov and Gromov-weak topology coincide 69
2.6. Ultra-metric measure spaces 70
2.7. Compact metric measure spaces 71
2.8. Distributions of random metric measure spaces 73
2.9. Equivalent metrics 75
Chapter 3. Examples of limit trees I: Branching trees 83
3.1. Excursions 83
3.2. The Brownian continuum random tree 87
3.3. Aldous’s line-breaking representation of the Brownian CRT 89
3
4 CONTENTS
3.4. Campbell measure facts: Functionals of the Brownian CRT 91
3.5. Existence of the reactant branching trees 99
3.6. Random evolutions: Proof of Theorem 3.5.3 103
Chapter 4. Examples of limit trees II: Coalescent trees 111
4.1. Λ-coalescent measure trees 112
4.2. Spatially structured Λ-coalescent trees 115
4.3. Scaling limit of spatial Λ-coalescent trees on Zd, d ≥ 3 120
4.4. Scaling limit of spatial Kingman coalescent trees on Z2 125
Chapter 5. Root growth and Regrafting 131
5.1. A deterministic construction 134
5.2. Introducing randomness 138
5.3. Connection to Aldous’s line-breaking construction 141
5.4. Recurrence, stationarity and ergodicity 145
5.5. Feller property 147
5.6. Asymptotics of the Aldous-Broder algorithm 156
5.7. An application: The Rayleigh process 158
Chapter 6. Subtree Prune and Regraft 163
6.1. A symmetric jump measure on (Twt,d ) 164
GHwt
6.2. Dirichlet forms 167
6.3. An associated Markov process 169
6.4. The trivial tree is essentially polar 172
Chapter 7. Tree-valued Fleming-Viot dynamics 181
7.1. The tree-valued Fleming-Viot martingale problem 181
7.2. Duality: A unique solution 186
7.3. Approximating tree-valued Moran dynamics 189
7.4. Compact containment: Limit dynamics exist 191
7.5. Limit dynamics are tree-valued Fleming-Viot dynamics 193
7.6. Limit dynamics yield continuous paths 200
7.7. Proof of the main results (Theorems 7.1.6 and 7.3.1) 201
7.8. Long-term behavior 202
7.9. The measure-valued Fleming-Viot process as a functional 203
7.10. More general resampling mechanisms and extensions 205
7.11. Application: Sample tree lengths distributions 208
Index 219
Bibliography 221
Many many thanks ...
Es fehlte mir weniger am Zutrauen zu promovieren.
Das w¨are der zweite Schritt vor dem ersten gewesen.
Christiane Leidinger1
Submitting my habilitation thesis is a good occasion to review the long
and often difficult way I have covered and, of course, to thank all of the peo-
ple who gave advice and supported me in relating me with the Habilitation
degree. Being aware of my social-cultural and educational background I go
beyond the common focus at the end of a habilitation project on the years
after the PhD and want to stress and acknowledge that it took the present
thesis each day of the last 35 years to finally become real.
Thanks and love to my mother Claudia Winter who has supported my
cravingforstudyingfrommyearliestdays,forexample,byfindingmepopu-
larscientificbooksinmathematicsandphysicsofwhichshehadonlyguessed
how much I appreciated them and for finding and defending a space of my
own in our much too small Prenzlauer Berg apartment. And to my sisters
JeanetteandSimoneWinterforrelievingmyfeelingsofbecomingastranger
to my family. I’m also very grateful to my father’s middle school teacher
and later friend of the family Brunhilde Lorff who guided me in feminism
more than 30 years ago.
I want to thank my teachers Elke Elsing, Frau Holl¨ander, Frau Matthes
and Frau Steinbrecher at the A¨nne and Anton Saefkow School who sup-
ported me in many aspects and who for the first time related me to the idea
of studying at a university - and even abroad - in the age of only 13. By
that time I had just joined a circle in the Mathematische Schu¨lergesellschaft
(MSG) Leonard Euler at the Humboldt University instructed by Ingmar
Lehman who introduced me to the academic way of approaching mathe-
matical problems and who believed in my abilities to an extend that I could
not ignore them anymore. Thanks to him and to all the people involved in
the organization of the yearly 10 days MSG summer camps which I hap-
pily remember as very challenging. And also to Gu¨nter Last who taught
me probability during my last year of high school and who advised my first
scientific thesis written as part of the requirements of my Abitur.
1KlasseN Dissertation U¨ber Arbeitert¨ochter und das Promovieren aus der Bildungs-
ferne, [Lei06]
5
6 MANY MANY THANKS ...
After I entered university - despite an optimum preparation - I lost self-
confidence with each day I was sitting in class and was too intimidated
to raise a single question and in the end often too confused to be able
to summarize what I had just learned. The strong feeling of alienation
often kept me from studying with my fellow students. It was only after 3
years of chosen isolation that I attended a seminar with Andreas Greven on
InteractingParticleSystemswhichallofasuddenexcitedandinspiredmein
manyways. IwanttothankUtaFreibergwhoalsoattendedtheseminarfor
letting me re-enjoy talking about mathematics. My great gratitude goes to
AndreasGreven,wholatersupervisedmyDiplomaandPhDthesesandwith
whomIhavecontinuedtoworkindifferentprojectssince. Heintroducedme
to the models arising in population genetics on which I seem to have built
a career now, he encouraged and supported me in traveling to international
conferences, but in the first place he adviced and emotionally supported me
over all the years. I also felt emotionally supported by my former colleague
and office mate Achim Klenke whom I would like to thank particularly for
sharing personal experiences at many occasions during my Diploma and
PhD theses.
SinceIhavestartedmyPhDIhavetraveledtomanyplaces. Theincom-
plete list of people whom I want to thank for invitations to research stays,
meetings and seminars include Siva Athreya, Ellen Baake, Don Dawson,
Frank den Hollander, Allison Etheridge, Steve Evans, Nina Gantert, An-
dreas Greven, Achim Klenke, Vlada Limic, Terry Lyons, Ed Perkins, Theo
Sturm, Silke Rolles, Jan Swart, Alain-Sol Sznitman, Anton Wakolbinger,
RuthWilliams, ... Ahugeimpacthaddefinitelymyfirstinternationalwork-
shop on Stochastic Partial Differential Equations and the attached summer
school at the University of British Colombia in Vancouver in 1997 at which
forthefirsttimeImetseveralofthepeoplewholaterbecamemyco-authors.
ThisjourneyaswellasaresearchstayoneandahalfyearslaterattheFields
InstituteinTorontoweremadepossibleduetoagrantofDonDawsonwho-
along with Anton Wakolbinger - later refereed my PhD thesis and has since
been following my track. I would like to thank both as well as Steve Evans
and Andreas Greven for writing letters of recommendation often with very
short notice. I would also like to thank for the additional financial support
from the Edith and Otto Haupt Foundation and the unconventional help of
Wolfgang Schmidt which made the trip to Vancouver possible.
It was one of my stays at the Fields Institute in Toronto from which I
broughtaposterwiththestrikingmessage“Gaysandlesbiansareourteach-
ers, students, parents, doctors, ...” which once hanging in the Mathematical
Institute in Erlangen created hot disputes. I would like to thank everybody
who was supportive during that time which was difficult for me, in particu-
lar, to Tanja Dierkes, Andreas Greven, Andreas Knauf, Peter Pfaffelhuber
and Iljana Z¨ahle who took a firm stand in the discussion. This is a welcome
opportunity to also thank all my gay and lesbian (to (may)be) colleagues
MANY MANY THANKS ... 7
at the department for coming out to me who would or would not like to see
their names written here.
The academic year 2002/03 I spent with a DFG research fellowship at
the University of California in Berkeley where I worked with Steve Evans
and Jim Pitman. I would like to thank them and all the people with whom
I interacted during that year. I am particularly grateful to Steve who intro-
duced me to the central theme of this thesis which is the space of real trees
and the Gromov-Hausdorff distance and from whom I learned a lot about
writing (and finishing) a paper. Working and specifically writing with him
is a great pleasure to me.
In the summer semester 2004 I did my first academic outing in the
humanities by attending a very inspiring reading seminar on “Written Iden-
tities”instructedbyDorisFeldmannattheEnglishDepartmentatErlangen
University. AsaconsequenceinFebruary2005-althoughstilltroubledwith
relating the mathematician that I am with lesbian-feminist research - I took
the courage to join the lfq network “Netzwerk lesbisch-feministisch-queerer
Forschung” initiated a year before by Christiane Leidinger.
Coming back from Berkeley to Erlangen I also started to take Hebrew
lessons, a bold venture in a region which inhabits not many native speakers.
However there is a class at the Bildungszentrum Nu¨rnberg which had been
taught for more than 10 years by Ganja Benari. Although, by the time
I joined, the class was far beyond my knowledge in Hebrew, I found in
Ganja and in all the women taking part in the course excellent teachers. So
many thanks to Batja, Dorothee, Edith, Ganja, Hiltrud, Margot, Renate,
Rosemarie, ... The present thesis was written during a current research stay
at the Technion in Haifa founded by the Aly Kaufman Foundation. I am
verygratefultoLeonidMytnikforinvitingandencouragingmetocomeand
to all the local probability people for their great hospitality. Thanks also
to Chen Weider who rented to me his wonderful sea view apartment which
had been my home for the last months and in which I enjoyed writing huge
parts of this thesis.
Iparticularlyacknowledgemyco-authorswhoseworkwithmeappearsin
thisthesis: StevenN.Evans, AndreasGreven, VladaLimic, PeterPfaffelhu-
ber, Jim Pitman and Lea Popovic as well as all my other collaborators Siva
Athreya, Michael Eckhoff, Janos Engl¨ander, Leonid Mytnik, Anja Sturm,
Rongfeng Sun and Iljana Z¨ahle. I am also thankful to all the anonymous
referees for the thorough reading of the papers. The revisions based on
their reports often improved the presentation. Further thanks to Michael
Eckhoff, Grit Paechnatz, Peter Pfaffelhuber, Ulrike Tisch and Iljana Z¨ahle
who kindly proof-read various parts of the manuscript.
Special thanks go to my house mates and friends Heike Herzog, Grit
Paechnatz and Kathrin Schmidt who have been with me through all the ups
and downs in the last years. And to the physiotherapists Perla Ben Simon,
Silke Kruse and Dorit Thu¨mer who professionally worked with me. I also
extend thanks for support that runs far beyond the bounds of collegiality to
8 MANY MANY THANKS ...
Lisa Beck, Nina Gantert, Vlada Limic, Lea Popovic and Iljana Z¨ahle. And
to Peter Pfaffelhuber from whom I learned how to collaborate with people
which may have different scholarly interests from my own.
An incomplete list of other colleagues and friends whom I would like to
thank for advise and support include David Aldous, Nihat Ay, Michaela
Baetz, Nadja Bennewitz, Dieter Binz, Juditha Cofman, Claudia Dem-
pel, Gabriele Dennert, Axel Ebinger, Silvia Eichner, Oye Felde, Simone
Fischer, Orit Furman, Walter Hofmann, Tobias J¨ager, Gerhard Keller,
Edith Kellinghusen, Julia Kempe, Manfred Kronz, Christiane Leidinger,
Alexander Lepke, Heike Lepke, Anna Levit, Wolfgang L¨ohr, Oded Regev,
Rosi Ringer, Gerhard Scheibel, Frank Schiller, Johanna Schmidt, Sarah
Schmiedel, Christoph Schumacher, J(.) Seipel, Thomas Springer, Ljiljana
Stamenkovic, Andrea Stroux, Sreekar Vadlamani, Stefanie Weigel, Silvia
Wendler, Yael Zbar, Helga Zech, ...
Anita Winter
Haifa, June 2007
Introduction
In the present thesis we study random trees and tree-valued Markov
dynamics which arise in the limit of discrete trees and discrete tree-valued
Markov chains, respectively, after a suitable rescaling, as the number of
vertices tends to ∞.
Random trees appear frequently in the mathematical literature. Promi-
nent examples are random binary search trees as a special case of ran-
dom recursive trees ([DH05]), ultra-metric structures in spin-glasses (see,
for example, [BK06, MPV87]), spanning trees (see, for example, [AS92,
KL96, PW98, BGL98]), ect. In branching models trees arise, for exam-
ple, as the Kallenberg tree and the Yule tree in the (sub-)critical or super-
critical, respectively, Galton-Watson process which is conditioned on “sur-
vival” ([Kal77, EO94]).
A huge enterprize in biology is phylogenetic analysis which reconstructs
the “family trees” of a collection of taxa. An introduction to mathematical
aspects of the subject are surveyed in [SS03]. Due to enormous diver-
sity in life phylogenetics often leads to the consideration of very large trees
and therefore demands for an investigation of limits of finite trees. For
example, by taking continuum (mass) limits we have for branching mod-
els the Brownian continuum random tree (Brownian CRT) or the Brow-
nian snake ([Ald91b, Ald93, LG99a]). More general branching mech-
anisms lead to more general genealogies, such as Levy trees which is the
infinite variance offspring distribution counterpart of the Brownian CRT
([DLG02]), the Poisson snake ([AS02]) or the reactant trees arising in cat-
alytic branching systems ([GPW06b]), to name just a few. In population
models with a fixed population size (or total mass), the genealogical trees
can be generated by coalescent processes, for example, the Kingman coa-
lescent tree ([Kin82a, Ald93, Eva00a]) or the Λ-coalescent measure trees
([GPW06a]).
Many results towards convergence of finite trees toward a limit tree
had been shown by considering the asymptotic behavior of functionals of
ensembles of random trees such as their height, total number of vertices,
averaged branching degree, ect. (see, for example, [CP00, Win02, PR04,
CKMR05]).
With a series of papers [Ald91a, Ald91b, Ald93] (see also [LG99a,
Pit02]) Aldous suggested a much stronger notion of convergence of random
trees. The main difficulty Aldous had to overcome was to map the sequence
9
10 INTRODUCTION
of trees into a space of all the “tree-like” objects which may arise in the
limit as the number of vertices tends to infinity. First, following a long
tradition, he relied on the connection between trees and continuous paths.
Encoding trees by continuous functions allows to think of weak convergence
of random trees as weak convergence of continuous functions with respect
to the uniform topology on compacta. In particular examples this approach
may have at least two drawbacks. Although there is a classical bijection
between rooted planar trees and lattice paths (see, for example, [DM00]),
there seems to be no obvious way to uniquely associate a continuous path
to a limit tree. Moreover the uniform topology is a rather strong topology.
Secondly, Aldous noticed that a finite leaf-labeled tree with edge lengths
is isomorphic to a compact subset of ℓ1, i.e., the space of non-negative sum-
mable sequences equipped with the Hausdorff topology. With this encoding
weak convergence of random trees translates into weak convergence of the
associated closed subsets of ℓ1 where the space of closed subsets of the met-
ric space ℓ1 is equipped, as usual, with the Hausdorff topology. Aldous’s
approach is extremely powerful. In particular, it allowed him to show that
a suitably rescaled family of Galton-Watson trees, conditioned to have total
population size n, converges as n → ∞ to the Brownian continuum random
tree, which can be thought of as the tree inside a standard Brownian excur-
sion. More recently his approach was applied in [HMPW] to identify the
self-similar fragmentation tree as the scaling limit of discrete fragmentation
trees and doing so to confirm in a strong way that the whole trees grow at
the same speed as the mean height of a randomly chosen leaf.
Both approaches via continuous paths and compact subsets of ℓ1 are
designed for leaf-labeled trees. If one wants to rescale unlabeled trees one
may be tempted to invent a labelling. Since the choice of a labelling playing
the role of “coordinates” is arbitrary, it may not always be handy to work in
this setting. One rather should be more consequent and follow an intrinsic
- that is, “coordinate free” - path.
Before we motivate this, we note that there is quite a large literature
on other approaches to “geometrizing” and “coordinatizing” spaces of trees.
The first construction of codes for labeled trees without edge-length goes
back to 1918: Pru¨fer ([Pru¨18]) sets up a bijection between labeled trees
of size n and the points of {1,2,...,n}n−2. Phylogenetic trees are identified
with points in matching polytopes in [DH98], and [BHV01a] equips the
space of finite phylogenetic trees with a fixed number of leaves with a metric
that makes it a cell-complex with non-positive curvature.
The thesis is split in three parts.
Part I: Real trees and metric measure spaces
Inthefirstpartofthepresentthesiswedevelopsystematicallythetopo-
logicalpropertiesofpossiblestatespacesandcharacterizethecorresponding
convergence in distribution.