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

Sujets

• Mathematics

Informations

• Publié par: universitat_bielefeld
• Publié le: 01 janvier 2011
• Nombre de lectures: 35
• Langue: English
• Poids de l'ouvrage: 1 Mo

Extrait

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) . . . . . . .