On the large time behavior of diffusions [Elektronische Ressource] : results between analysis and probability / Martin Kolb

Sujets

• Mathematics

Informations

• Publié par: technische_universitat_kaiserslautern
• Publié le: 01 janvier 2009
• Nombre de lectures: 22
• Langue: English

Extrait

Vom Fachbereich Mathematik der Technischen Universit¨at Kaiserslautern zur Verleihung
des akademischen Grades Doktor der Naturwissenschaften (Dr. rer. nat.) genehmigte
Dissertation
On the Large Time Behavior of Diffusions
Results Between Analysis and Probability
MartinKolb
¨1. Gutachter: Prof. Dr. Heinrich von Weizsacker
2. Gutachter: Prof. Dr. Carl Mueller
Vollzug der Promotion: 29.10.2009
D 386iiContents
1 Introduction 1
1.1 Demographic Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2 Quasistationary Distributions in the Regular Case 9
2.1 Assumptions, Definitions and Previous Results . . . . . . . . . . . . . . . . 9
2.2 Analytic Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.3 Convergence to Quasistationarity . . . . . . . . . . . . . . . . . . . . . . . 30
2.3.1 0 Regular and∞ Natural . . . . . . . . . . . . . . . . . . . . . . . 31
2.3.2 The Case of an Entrance Boundary at Infinity . . . . . . . . . . . . 46
2.3.3 Concluding Remarks and Open Problems . . . . . . . . . . . . . . . 47
3 Quasistationary Distributions : the Non-regular Case 49
3.1 One-dimensional Diffusions on the half-line . . . . . . . . . . . . . . . . . . 50
3.2 Spectral decomposition of L . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.3 Existence of Quasistationary Distributions . . . . . . . . . . . . . . . . . . 65
3.4 Concluding Remarks and Open Problems . . . . . . . . . . . . . . . . . . . 76
4 Super-Brownian Motion with a Single Point Source 79
4.1 Analytic Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
4.1.1 Point Interactions in Three Dimensions . . . . . . . . . . . . . . . . 82
4.1.2 Point Intera in Two Dimensions . . . . . . . . . . . . . . . . . 83
4.2 Super-Brownian Motion with a Single Point Source . . . . . . . . . . . . . 90
4.3 Law of Large Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
4.3.1 Scaling of the Expectation . . . . . . . . . . . . . . . . . . . . . . . 92
4.3.2 Pathwise Large Time Behavior . . . . . . . . . . . . . . . . . . . . 99
4.4 Concluding Remarks and Open Problems . . . . . . . . . . . . . . . . . . . 108
5 Selfadjointness of Schr¨odinger operators 111
5.1 The Main Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
5.2 Proof of Theorem 5.1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
iiiiv CONTENTSPreface
This thesis contains results which have been established during my Phd-studies under su-
pervision of Prof. Dr. Heinrich von Weizs¨acker. I was deeply impressed by his broad
interest and mathematical knowledge. This work would definitely not have been possible
without the stimulating research environment provided by the research group led by Prof.
Dr. Heinrich von Weizsa¨cker.
Thisthesisdoesnotcontainallresultswhichhavebeenobtainedduringthelastyears. The
interested reader will find further results in the domain of intersection of probability and
analysis in [1] and [58]. Included are mainly those results, which really reflect the influence
of the ’Kaiserslautern school of probability’. The investigation of processes conditioned
on unlikely events has been a recurring theme in Kaiserslautern. In this thesis we aim to
continue this tradition.
Several outcomes concerning quasistationary distributions for one-dimensional diffusions
result from very stimulating discussions with Dr. David Steinsaltz during a pleasent stay
at Oxford University (Worcester college). Furthermore, some inspiring results of his col-
laborationwithProf. Dr. SteveEvansinitiatedmypresentjointinvestigationwithRobert
Grummt of an extremely interesting super-process, which has been constructed by Fleis-
chmann and Mueller.
At this point I would like to take the opportunity to thank Prof. Dr. Carl Mueller
for his joint work with Klaus Fleischmann and for serving as external referee for this work.
Moreover it is a great pleasure to thank Prof. Dr. Fritz Gesztesy for patiently answering
several questions concerning his joint work with Dr. Maxim Zinchenko. Moreover I am
particularly grateful to him for having read and discussed the analytic part of Chapter 3.
It is planned to submit several results of this work for publication in part jointly with
Dr. David Steinsaltz, Leif D¨oring and Robert Grummt, respectively. The main ideas and
results presented in this work are due to myself.
vvi CONTENTSCurriculum Vitae
Personal Data
• Birthdate: Januar 06, 1980
• Citizenship: german
Education
09/1986 –06/1999 Karlsgymnasium Munchen-P¨ asing
10/1999 – 07/2000 civil service at the hospital ’Kreisklinikum Furstenfeldbr¨ uck’
10/2000 –08/2002 Study of mathematics, logic and theory of science
at the Ludwig-Maximilians-Universit¨at in Munich
08/2002–07/2003 Study of mathematics at the University of Copenhagen
08/2003 –01/2005 Study of mathematics, logic and theory of science
at theat in Munich
Employment
02/2005–08/2005 ’wissenschaftlicher Mitarbeiter’ at the University of Constance
09/2005–08/2008 Phd-studies at the University of Kaiserslautern
09/2008–31.12.2009 ’wissenschaftlicher Mitarbeiter’ at the
Ludwig-Maximilans Universit¨at Munich
Publications
• On the strong uniqueness of some finite dimensional Dirichlet operators. Infin. Dimens.
Anal. Quantum Probab. Relat. Top. 11 (2008), no. 2, 279–293.
• (with A. Bassi and D. Dur¨ r)On the tong time behavior of free stochastic Schr¨odinger
Evolutions, Rev. Math. Phys. 22 (2010), 55–89
viiviii CONTENTSChapter 1
Introduction
The study of quasistationary distributions is a long standing problem in several areas of
probability theory and a complete understanding of the structure of quasistationary dis-
tributions seems to be available only in rather special situations such as Markov chains on
finite sets or more general processes with compact state space. For a regularly updated
extensive bibliography with about 380 entries concerning the topic of quasistationary dis-
tributions we refer to [77]. In this work we consider one-dimensional diffusions on the
half-line and study the problem of convergence to quasistationarity. The starting point of
our investigation is the recent contribution [84] of S. Evans and D. Steinsaltz. It might
be rather surprising that despite the key contributions [25], [65], [66], [84] and [20] the
structure of quasistationary distribution of one-dimensional killed diffusions has not been
completely clarified. Even worse, since the proof of the main result of [25] has a serious
1gap even for the case of one-dimensional diffusions with trivial internal killing the general
picture is still rather incomplete. Similar problems for other classes of stochastic processes
have also been investigated quite frequently (see e.g. the important contributions [41] and
[56] of H. Kesten and his co-authors and the work [89] of E. van Doorn, where classes of
Markov-chains onthe integers andin particularbirthanddeath chains are considered) but
often a complete understanding is still missing.
ThefirstworkconcerningtheYaglomlimitforspecialone-dimensionaldiffusionsincluding
non-trivial internal killing seems to be [54]. Some results for one-dimensional diffusions
with a compact state space have been established by N. Sidorova in chapter 4 of her phd-
thesis under supervision of Prof. H. von Weizs¨acker (see also [80]). The case of general
one-dimensional diffusions with a non-compact state space seems to be considered for the
first time in [84]. Steinsaltz and Evans establish in [84] several results concerning the qua-
sistationary convergence of one-dimensional diffusions killed at the boundary and in the
interior of the state space. In particular they prove an interesting dichotomy. Under quite
general conditions a one-dimensional diffusion conditioned on long survival either runs off
to infinity or converges to a quasistationary distribution given by the lowest eigenfunction
1J.SanMart´ıncommunicatedtotheauthorseveralnewideas, whichmightafterprovidingsomefurther
arguments finally lead to a rigorous new proof of the results in [25]. Moreover we should stress that the
ideas developed in [25] played an important role in further developments.
16
6
2 CHAPTER 1. INTRODUCTION
of the generator. In this thesis we complete some of the results of Steinsaltz and Evans by
giving conditions which allow to decide whether convergence to quasistationarity or escape
to infinity occurs. Furthermore, we will be able to remove some unnecessary conditions
posed in [84]. Unfortunately there are still several natural questions, which we leave open.
These are collected at the end of each chapter.
ThedichotomyofEvansandSteinsaltzisderivedmainlybypurelyprobabilisticarguments.
IncontrasttothisweincludeseveralbasicfactsfromtheanalytictheoryofSturm-Liouville
operators. Although it is well-known since the pioneering work of Mandl [63] that the
convergence to quasistationary distributions for one-dimensional diffusions is intimately
connected to the bottom of the spectrum of the diffusion generator, only elementary rudi-
ments and in many proceeding works on the problem nothing more than definitions and
verybasicresultsoftherichspectraltheoryhavebeenused. Theinclusionofsomeanalytic
methods allows to provide a more transparent picture. The use of these techniques even
seems to be necessary s