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Stochastic analysis related to Gamma measures [Elektronische Ressource] : Gibbs perturbations and associated diffusions / Dennis Hagedorn. Fakultät für Mathematik

250 pages
ThesisStochastic Analysis related to Gammameasures- Gibbs perturbations and associated DiffusionsDissertationzurErlangung des Doktorgrades (Dr. math.)derFakultät für MathematikderUniversität Bielefeldvorgelegt vonDennis Hagedorn aus Minden (Westf.)imNovember 2011iiIm Zuge der Veröffentlichung wurde die vorliegende Dissertation redaktionellkorrigiert.Bielefeld, im Dezember 2011iiiSummaryIn this thesis (consisting of Parts I - III) we study Gamma measures located on thedcone (R ) of discrete Radon measures. They form, as well as the Gaussian andPoisson measures, an important class of measures on infinite dimensional spacesand appeared in the representation theory of groups. In the present thesis, thefollowing topics of Gamma analysis are developed: Construction of Gibbs perturbations for the Gamma measuresd Differential structure on the cone (R ) Integration by parts formulas for Gamma and Gibbs measures Construction of associated diffusionsdIn Part I, we define a homeomorphism between the cone (R ) and a subsetd d^ ^of the configuration space ( R ) over the product space R of marks in R :=+d(0;1) and positions in R . This subset consists of pinpointing configurationsdwith finite local mass. Then we construct Gamma measures on (R ) as imaged^measures, under , of proper Poisson measures on ( R ).In Part II, we establish Gibbs perturbations of Gamma measures w.r.t.
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Thesis
Stochastic Analysis related to Gamma
measures
- Gibbs perturbations and associated Diffusions
Dissertation
zur
Erlangung des Doktorgrades (Dr. math.)
der
Fakultät für Mathematik
der
Universität Bielefeld
vorgelegt von
Dennis Hagedorn aus Minden (Westf.)
im
November 2011ii
Im Zuge der Veröffentlichung wurde die vorliegende Dissertation redaktionell
korrigiert.
Bielefeld, im Dezember 2011iii
Summary
In this thesis (consisting of Parts I - III) we study Gamma measures located on the
dcone (R ) of discrete Radon measures. They form, as well as the Gaussian and
Poisson measures, an important class of measures on infinite dimensional spaces
and appeared in the representation theory of groups. In the present thesis, the
following topics of Gamma analysis are developed:
Construction of Gibbs perturbations for the Gamma measures
d Differential structure on the cone (R )
Integration by parts formulas for Gamma and Gibbs measures
Construction of associated diffusions
dIn Part I, we define a homeomorphism between the cone (R ) and a subset
d d^ ^of the configuration space ( R ) over the product space R of marks in R :=+
d(0;1) and positions in R . This subset consists of pinpointing configurations
dwith finite local mass. Then we construct Gamma measures on (R ) as image
d^measures, under , of proper Poisson measures on ( R ).
In Part II, we establish Gibbs perturbations of Gamma measures w.r.t. a pair
potential that describes the interaction of particles and satisfies certain stability
properties: We follow the Dobrushin-Lanford-Ruelle approach to Gibbs random
fields in classical statistical mechanics and introduce the corresponding Gibbs for-
malism on the cone. Proving the existence of the Gibbs measures on the cone
d(R ) is a non-trivial problem, even for a non-negative potential. We know about
d d^the cone (R ) less than about the configuration space ( R ), hence we transfer
d 1 d^ ^the problem to ( R ) via the homeomorphism . Even on ( R ), the trans-
fered potential with infinite range does not fit the standard framework because of
the high concentration close to 0 of the underlying intensity measure on R . We+
develop analytic techniques, involving Lyapunov functionals and weak dependence
d^on boundary conditions, to construct Gibbs measures on ( R ) and characterize
sets supporting them. Using the homeomorphism , we establish the existence of
Gibbs perturbations on the cone.
To obtain diffusions on the cone, in Part III, we introduce a gradient which
consists of extrinsic and intrinsic parts. They correspond to the motion of marks
andpositionsofparticles,respectively. Animportantresulthere(andanewissuein
infinite dimensions) is an integration by parts formula without an underlying quasi-
invariance property of the involved Gamma measure. Next, we study conservative
gradient Dirichlet forms of Gibbs measures constructed in Part II. To check their
quasi-regularity, we define a Polish space, in which we embed the cone. Therefore,
we study a priori diffusions on the Polish space. A crucial issue here is that the
1 ddiffusions are actually located on a subset of ( (R )). Using this fact and the
homeomorphism , we construct diffusions on the cone. In particular, we get an
example of diffusions describing the motion of densely distributed particles.
KKKKTKTKKTTTTivContents
1 Introduction 1
1.1 Infinite dimensional analysis . . . . . . . . . . . . . . . . . . . 3
1.1.1 Gaussian analysis . . . . . . . . . . . . . . . . . . . . . 4
1.1.2 Poisson measure . . . . . . . . . . . . . . . . . . . . . . 6
1.1.3 Gamma . . . . . . . . . . . . . . . . . . . . . 10
1.2 Content . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.2.1 Gamma measures . . . . . . . . . . . . . . . . . . . . . 11
1.2.2 Gibbs perturbations . . . . . . . . . . . . . . . . . . . 13
d1.2.3 Differential calculus over (R ) . . . . . . . . . . . . . 18
1.3 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . 24
I Gamma and Gamma-Poisson measures 25
2 Poisson measures 27
2.1 A short introduction to configuration spaces . . . . . . . . . . 28
2.1.1 Configuration space . . . . . . . . . . . . . . . . . . . 29
2.1.2 Poisson measure . . . . . . . . . . . . . . . . . . . . . . 31
2.2 Gamma-Poisson measures . . . . . . . . . . . . . . . . . . . . 33
^2.2.1 Poisson measure on ( X) . . . . . . . . . . . . . . . . 34
3 Gamma measures 39
3.1 Levy . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.1.1 The cone (X) . . . . . . . . . . . . . . . . . . . . . . 40
3.1.2 Levy measures . . . . . . . . . . . . . . . . . . . . . . 41
3.2 Gamma measures . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.2.1 Gamma measuresG . . . . . . . . . . . . . . . . . . . 42
3.2.2 Moments of Gamma measures . . . . . . . . . . . . . . 44
3.2.3 Multiplicative Lebesgue measureL . . . . . . . . . . . 45
add3.2.4 Additive Lebesgue measureL . . . . . . . . . . . . . 46
add3.3 Basic properties ofG ,L andL . . . . . . . . . . . . . . . 46
v
KKvi CONTENTS
3.3.1 Quasi-invariance, ergodicity and extremality ofG . . . 46
3.3.2 Projective invariance and convex combinations ofL
addandL . . . . . . . . . . . . . . . . . . . . . . . . . . 48
II Gibbs perturbations 51
4 Gibbs measures with non-negative potentials 57
^4.1 Gibbsian formalism on ( X) . . . . . . . . . . . . . . . . . . . 59
4.1.1 Potential . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.1.2 Relative energy . . . . . . . . . . . . . . . . . . . . . . 60
4.1.3 Local specification . . . . . . . . . . . . . . . . . . . . 61
4.1.4 Gibbs measures . . . . . . . . . . . . . . . . . . . . . . 63
4.2 Existence of Gibbs measures: Basic model . . . . . . . . . . . 64
4.2.1 Support of the local specification kernels . . . . . . . . 67
4.2.2 Local equicontinuity . . . . . . . . . . . . . . . . . . . 70
4.2.3 Existence of Gibbs measures: Basic model . . . . . . . 72
^4.3 Existence of Gibbs measures on ( X): General case . . . . . . 75
4.3.1 A (general) local (w.r.t.
m) mass map . . . . . . . 76
4.3.2 Support of the Gamma-Poisson measure . . . . . . . . 79
4.3.3 Finiteness of the relative energy . . . . . . . . . . . . . 80
4.3.4 Support of the local specification kernels . . . . . . . . 81
4.3.5 Local equicontinuity . . . . . . . . . . . . . . . . . . . 84
4.3.6 Existence . . . . . . . . . . . . . . . . . . . . . . . . . 84
4.3.7 Support of Gibbs measures . . . . . . . . . . . . . . . . 90
4.3.8 Higher moments of Gibbs measures . . . . . . . . . . . 92
4.4 A closer look at Gibbs measures . . . . . . . . . . . . . . . . . 93
d^4.4.1 Covering ofR . . . . . . . . . . . . . . . . . . . . . . 94
4.4.2 An associated global mass . . . . . . . . . . . . . . . . 96
4.4.3 Support properties of Gibbs measures . . . . . . . . . . 101
4.4.4 Compactness of the set of Gibbs measure . . . . . . . . 102
^4.5 A modified description of Gibbs measures on ( X) . . . . . . 103
4.5.1 Semi-local specification . . . . . . . . . . . . . . . . . . 104
4.5.2 A modified concept of Gibbs measures . . . . . . . . . 108
5 Gibbsian measure for general potentials 111
^5.1 Gibbsian formalism on ( X) . . . . . . . . . . . . . . . . . . . 112
5.1.1 The potential in the basic model . . . . . . . . . . . . 112
^5.1.2 Partition of the space X . . . . . . . . . . . . . . . . . 113
5.1.3 A potential V in the general framework . . . . . . . . . 115
5.1.4 Gibbsian formalism . . . . . . . . . . . . . . . . . . . . 118CONTENTS vii
5.2 Existence for general potentials . . . . . . . . . . . . . . . . . 124
5.2.1 Weak dependence on boundary conditions . . . . . . . 126
5.2.2 Uniform bounds for local Gibbs states . . . . . . . . . 129
5.2.3 Local equicontinuity . . . . . . . . . . . . . . . . . . . 130
5.2.4 Existence of Gibbs measures . . . . . . . . . . . . . . . 130
5.2.5 Moment estimates for Gibbs measures . . . . . . . . . 132
5.3 Gibbs measures on the cone . . . . . . . . . . . . . . . . . . . 134
5.3.1 Gibbsian formalism on (X) . . . . . . . . . . . . . . 134
5.3.2 One-to-onecorrespondencebetweenbetweenGibbsmea-
^sures on (X) and ( X) . . . . . . . . . . . . . . . . . 137
5.3.3 Existence of Gibbs measures . . . . . . . . . . . . . . . 138
III Differential calculus 141
6 Differential calculus and Dirichlet forms 143
6.1 Differential geometry on the cone (X) . . . . . . . . . . . . 145
6.1.1 Group of motions . . . . . . . . . . . . . . . . . . . . . 146
6.1.2 Extrinsic Gradient . . . . . . . . . . . . . . . . . . . . 146
6.1.3 Intrinsict . . . . . . . . . . . . . . . . . . . . . 149
6.1.4 Joint gradient . . . . . . . . . . . . . . . . . . . . . . . 151
26.2 Dense subsets of L ( (X);G ) . . . . . . . . . . . . . . . . . . 152
6.2.1 A set of point separating functions . . . . . . . . . . . 153
6.2.2 Denseness criterium . . . . . . . . . . . . . . . . . . . . 157
6.3 Integration by parts and Dirichlet forms . . . . . . . . . . . . 160
6.3.1 Intrinsic motion for Levy measures . . . . . . . . . . . 160
6.3.2 Extrinsic related to Gibbs measures . . . . . . 167
6.3.3 Intrinsic motion to Gibbs . . . . . . . 172
6.3.4 Joint motion related to Gibbs measures . . . . . . . . . 177
7 Equilibrium processes 181
7.1 An extrinsic process moving finite measures . . . . . . . . . . 184
7.1.1 Embedding of (X) . . . . . . . . . . . . . . . . . . . 184
7.1.2 Quasi-regularity of extrinsic Dirichlet forms . . . . . . 185
7.2 Quasi-regularity on multiple configurations . . . . . . . . . . . 188
7.2.1 The embedding space for compact X . . . . . . . . . . 188
d7.2.2 Identifying a proper Polish space for X =R . . . . . . 192
^7.2.3 Bilinear forms on ( X) . . . . . . . . . . . . . . . . . . 193
7.2.4 Closability of the bilinear forms . . . . . . . . . . . . . 196
7.2.5 Quasi-regularity for compact X . . . . . . . . . . . . . 199
d7.2.6y for X =R . . . . . . . . . . . . . . . 206
KKKKKviii CONTENTS
7.3 Diffusions on multiple configurations . . . . . . . . . . . . . . 208
7.3.1 Diffusions on multiple configurations . . . . . . . . . . 209
7.3.2 Exceptional set . . . . . . . . . . . . . . . . . . . . . . 210
7.4 Diffusions on the cone . . . . . . . . . . . . . . . . . . . . . . 214
d7.4.1 Diffusions on (R ) . . . . . . . . . . . . . . . . . . . 214
d7.4.2 Extension of Dirichlet forms on (R ) . . . . . . . . . 217
A Spaces of measures 221
A.1 Kuratowski’s theorem . . . . . . . . . . . . . . . . . . . . . . . 221
A.2 Properties of Radon measures . . . . . . . . . . . . . . . . . . 222
IV Bibliography and Index 227
Bibliography 229
Indexes 237
List of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238
KKChapter 1
Introduction
Handling and modeling complex systems have become an essential part of
modern science. For a long time, complex systems have been treated in
physics, where e.g. methods of probability theory are used to determine their
macroscopic behavior by their microscopic properties. Nowadays complex
systems, ranging from e.g. ecosystems to the climate, biological populations,
societies and financial markets, play an important role in various fields such
as biology, chemistry, robotics, computer science and even social science.
A mathematical tool to study these systems is infinite dimensional analy-
sis. Widely applied, e.g. in financial mathematics and mathematical physics,
are Gaussian and Poissonian analysis corresponding to Gauss, resp. Poisson
measures. We develop some Gamma analysis related to Gamma measures,
which may serve to model biological systems.
The mentioned measures are infinite dimensional analogues of measures
classified by Meixner. A first step in the related analysis is to study sets
supporting them. In particular, Gaussian measures are located on linear
spaces; Poisson measures are supported by the space of locally finite con-
figurations. And Gamma measures have full mass on the cone of locally
finite, discrete Radon measures. The analysis developed for the Gaussian
and Poissonian measures includes chaos decompositions, differential struc-
tures on the underlying spaces, corresponding Dirichlet forms and associated
diffusions, whereas for the Gamma measures a chaos decomposition and a
quasi-invariancepropertyw.r.t. multiplicationofmarksisknown. Oneofour
aims is to introduce a differential structure on the cone, construct Dirichlet
forms and get associated diffusions on the cone.
An important feature of complex systems is the interaction of their com-
ponents. Let us exemplify this with a prominent physical example, namely
a gas: To model a free gas, Poisson measures are used. ’Free’ means that
12 CHAPTER 1. INTRODUCTION
any interaction of the molecules is absent. But, the molecules of real gases
interact with each other. To model this, the notion of Gibbs perturbations
of Poisson measures has been introduced and studied. Considering Gamma
measure as states of free systems, we will also study Gibbs perturbations of
Gamma measures.
A mathematical model for the above mentioned many-particle systems,
namely spaces of locally finite configurations, appeared first in statistical
mechanics. Such a configuration space describes the positions of identical
dparticles in a phase space, e.g. R . Here, locally finite means that there are
only finitely many particles in any compact area. To describe the allocation
of particles a Poisson measure with a certain intensity measure can be used.
It distributes the particles independently of each other (cf. e.g. Subsection
1.1.2 or Chapter 2).
As mentioned above, particles may interact and influence each other.
Gibbs measures are suitable to describe this phenomena. In the late 1960s
Dobrushin,LanfordandRuelleintroducedthemathematicalsettingforGibbs
measures that are used to describe equilibrium states of infinitely large sys-
tems (cf. [Dob68, Dob70b, LR69, Rue69]), which strongly encouraged the
development of the theory of Markov random fields (cf. [Geo88, Pre76]).
Generally speaking, one distinguishes between two main classes of Gibbs
measures, namely, spin systems on graphs or discrete metric spaces (cf., e.g.,
d[Lan20, Isi25, Geo88]) and particle systems in continuum, e.g., in R (cf.,
e.g., [AKR98b, Kun99, AKPR06]).
dWe will treat Gibbs measures for particle systems in the continuum R ,
d2N, and an attached space of marksR := [0;1) with an infinite measure+
on the marks. Thus, we extend models treated in the second class.
Standard references for the theory of Gibbs measures are [Geo88, Pre76].
More recent ones are [AKPR06] (an overview) and [KPR10] (an analytic
approach).
Aims
We will develop some structure of the Gamma analysis (cf. also Section 1.2):
We study Gibbs perturbations of the Gamma measures,
introduce a differential structure on the cone of discrete measures,
establish integration by parts formulas and
construct diffusions corresponding to associated Dirichlet forms.

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