Hyperbolic Dynamics and Brownian Motion: An Introduction

309 Pages·2012·1.597 MB·English

by  Jacques Franchi, Yves Le Jan

Hyperbolic Dynamics
and
Brownian Motions
an introduction
Jacques FRANCHI and Yves LE JAN
January 2011
2
CONTENTS
Introduction page 5
Summary page 6
I) The Lorentz-Mo¨bius group page 9
I.1) Lie algebras and groups: elementary introduction page 9
I.2) The Minkowski space and pseudo-metric page 18
I.3) The Lorentz-Mo¨bius group and its Lie algebra page 20
I.4) Two remarkable subgroups of PSO(1,d) page 23
I.5) Structure of the elements of PSO(1,d) page 26
I.6) The hyperbolic space Hd and its boundary ∂Hd page 30
I.7) Cartan and Iwasawa decompositions of PSO(1,d) page 32
II) Hyperbolic Geometry page 37
II.1) Geodesics and Light Rays page 37
II.2) A commutation relation page 46
II.3) Flows and leaves page 48
II.4) Structure of horospheres, and Busemann function page 52
III) Operators and Measures page 59
III.1) Casimir operator on PSO(1,d) page 59
III.2) Laplace operator page 61
D
III.3) Haar measure of PSO(1,d) page 64
Sd−1
III.4) The spherical Laplacian ∆ page 73
III.5) The hyperbolic Laplacian ∆ page 76
III.6) Harmonic, Liouville and volume measures page 80
IV) Kleinian groups page 91
IV.1) Terminology page 91
IV.2) Dirichlet polyhedrons page 93
IV.3) Parabolic tesselation by an ideal 2n-gone page 94
IV.4) Examples of modular groups page 101
3
IV.5) 3-dimensional examples page 116
V) Measures and flows on Γ Fd page 119
\
V.1) Measures of Γ-invariant sets page 119
V.2) Ergodicity page 121
V.3) A mixing theorem page 124
V.4) Poincar´e inequality page 129
VI) Basic Itˆo Calculus page 143
VI.1) Discrete martingales and stochastic integrals page 143
VI.2) Brownian Motion page 147
VI.3) Martingales in continuous time page 149
VI.4) The Itoˆ integral page 154
VI.5) Itoˆ’s Formula page 160
VI.6) Stratonovitch integral page 171
VII) Linear S. D. E.’s and B. M. on groups of matrices page 173
VII.1) Stochastic Differential Equations page 173
VII.2) Linear Stochastic Differential Equations page 178
VII.3) Approximation of left B.M. by exponentials page 192
VII.4) Lyapounov exponents page 199
VII.5) Diffusion processes page 201
VII.6) Examples of group-valued Brownian motions page 204
VIII) Central Limit Theorem for geodesics page 227
VIII.1) Adjoint Pd-valued left Brownian motions page 228
VIII.2) Two dual diffusions page 234
VIII.3) Spectral gap along the foliation page 236
VIII.4) Resolvent kernel and conjugate functions page 242
VIII.5) Contour deformation page 245
VIII.6) Divergence of ωf page 249
VIII.7) Sina¨ı’s Central Limit Theorem page 255
4
IX) Appendix relating to geometry page 267
IX.1) Structure of symmetrical tensors in R1,d page 267
IX.2) Another commutation relation in PSO(1,d) page 272
IX.3) The d’Alembertian (cid:50) on R1,d page 277
IX.5) Core-cusps decomposition page 282
X) Appendix relating to stochastic calculus page 287
X.1) A simple construction of real Brownian Motion page 287
X.2) Stochastic Riemanniann sums page 290
X.3) Chaos expansion page 292
XI) Index of notations, terms, and figures page 295
XI.1) General notations page 295
XI.2) Other notations page 296
XI.3) Index of Terms page 298
XI.4) Table of Figures page 304
References page 305
5
Introduction
This book provides first an elementary introduction to hyperbolic geometry, based
on the Lorentz group. Secondly, it introduces the hyperbolic Brownian motion and
related diffusions on the Lorentz group. Thirdly, an analysis of the chaotic behaviour
of the geodesic flow is performed using stochastic analysis methods. The main result
is Sina¨ı’s central limit theorem.
These methods had been exposed some years ago in research articles addressed to
experienced readers. In this book the necessary material of group theory and stochastic
analysis is exposed in a self-contained and voluntarily elementary way.
Only basic knowledge of linear algebra, calculus and probability theory is required.
Of course the reader familiar with hyperbolic geometry will traverse rapidly the
first five chapters. Those who know stochastic analysis will do the same with the sixth
chapter and the beginning of the seventh one.
Our approach of hyperbolic geometry is based on special relativity. The key role is
played by the Lorentz-M¨obius group PSO(1,d), Iwasawa decomposition, commutation
relations, Haar measure, and the hyperbolic Laplacian.
Thereisalotofgoodexpositionsofstochasticanalysis. Wetriedtomakeitasshort
and elementary as possible, to the purpose of making it easily available to analysts and
geometers who could legitimately be reluctant to have to go through fifty pages before
getting to the heart of the subject. Our exposition is closer to Itoˆ’s and McKean’s
original work (see [I], [MK]).
Themainresultsandproofs(atleastinthecontextofthisbook)areprintedinlarge
font. The reader may at first glance through the remaining part, printed in smaller
font.
Finally, some related results, which are never used to prove the main results, but
complete the expositions of stochastic calculus and hyperbolic geometry, are given in
the appendix. For the sake of completeness, the appendix also contains a construction
of Brownian motion.
6
Summary
I. ThefirstchapterdealswiththeLorentzgroupPSO(1,d),whichisthe(connected
component of the unit in the) linear isometry group of Minkowski space-time. It is
isomorphic to the Mo¨bius group of direct hyperbolic isometries.
It begins with an elementary and short introduction to Lie algebras of matrices and as-
sociated groups. Then the Minkowski space R1,d and its pseudo-metric are introduced,
together with the Lorentz-Mo¨bius group, and the space Fd of Lorentz frames, on which
PSO(1,d) acts both on the right and on the left.
We introduce then a subgroup Pd, generated by the first boost and the parabolic
translations, and we determine the conjugation classes of PSO(1,d).
The hyperbolic space Hd is defined as the unit pseudo-sphere of the Minkowski space
R1,d. Iwasawa’s decomposition of PSO(1,d) is given, and yields Poincar´e coordinates
in Hd. While the hyperbolic metric relates to Cartan’s decomposition of PSO(1,d).
II. The second chapter presents basic notions of hyperbolic geometry: geodesics,
light rays, tangent bundle, etc. Then the geodesic and horocycle flows are defined, by
a right action of Pd on frames. Horospheres and Busemann function are presented.
III. The third chapter deals with operators and measures. The Casimir operator
Ξ on PSO(1,d) induces the Laplace operator on Pd, and the hyperbolic Laplacian
D
∆. The Haar measure of PSO(1,d) is determined and shown to be bilateral. This
chapter ends with the presentation of harmonic, Liouville, and volume measures, and
their analytical expressions.
IV. The fourth chapter deals with the geometric theory of Kleinian groups and
their fundamental domains. It begins with the example of the parabolic tesselation of
the hyperbolic plane by means of an ideal 2n-gone. Then Dirichlet polyhedrons and
modular groups are discussed, with Γ(2) and Γ(1) as main examples.
V. In the fifth chapter we consider measures of Γ-invariant sets, and establish a
mixing theorem. We derive a Poincar´e inequality for the fundamental domain of a
generic geometrically finite, cofinite Kleinian group, i.e. a spectral gap for the corre-
sponding Laplacian.
VI. The sixth chapter deals with the basic Itoˆ calculus (Itˆo integral and for-
mula), starting with a short account of the necessary background about martingales
and Brownian motion.
VII. The seventh chapter construction of (left and right) Brownian motions on
groups of matrices, as solutions to linear stochastic differential equations. We establish
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in particular that the solution of such an equation lives in the subgroup associated to
the Lie subalgebra generated by the coefficients of the equation. Then we concentrate
on important examples: the Heisenberg group, PSL(2), Pd and the Poincar´e group
d+1. We also introduce basic stochastic analysis on matrices, and use it to define the
P
spherical and hyperbolic Brownian motions by means of a projection.
VIII. In the eighth chapter we provide a proof of the Sina¨ı Central Limit Theo-
rem, generalised to the case of a geometrically finite and cofinite Kleinian group. This
theorem indicates that asymptotically geodesics behave chaotically, and yields a quan-
titative expression of this phenomenon. The method we use is by establishing such
result first for Brownian trajectories, which is easier because of their strong indepen-
dence properties. Then we compare geodesics with Brownian trajectories, by means
of a change of contour. This requires in particular to consider diffusion paths on the
stable foliation and to derive the existence of a key potential kernel from the spectral
gap exhibited in Chapter V.
8
Chapter I
The Lorentz-Mo¨bius group
A large part of this book is centred on a careful analysis of this crucial
group, to which this first chapter is mainly devoted.
I.1 Lie algebras and groups: introduction
I.1.1 (d) and Lie subalgebras of (d)
M M
We shall here consider only algebras and groups of matrices, that is,
subalgebras of the basic Lie algebra (d), the set of all d d real square
M ×
matrices (for some integer d 2), and subgroups of the basic Lie group
≥
GL(d), the set of all d d real square invertible matrices, known as the
×
general linear group.
The real vector space ( (d),+, ) is made into an algebra, called a
M ·
Lie algebra, by means of the Lie bracket
[M,M ] := MM M M = [M ,M],
(cid:48) (cid:48) (cid:48) (cid:48)
− −
MM being the usual product of the square matrices M and M . The Lie
(cid:48) (cid:48)
bracket satisfies clearly the Jacobi identity: for any matrices M,M ,M
(cid:48) (cid:48)(cid:48)
we have
[[M,M ],M ] + [[M ,M ],M] + [[M ,M],M ] = 0.
(cid:48) (cid:48)(cid:48) (cid:48) (cid:48)(cid:48) (cid:48)(cid:48) (cid:48)
9
¨
10 CHAPTER I. THE LORENTZ-MOBIUS GROUP
The adjoint action of (d) on itself is defined by:
M
ad(M)(M ) := [M,M ], for any M,M (d).
(cid:48) (cid:48) (cid:48)
∈ M
The Jacobi identity can be written as follows: for any M,M ,
(cid:48)
∈ G
(cid:0) (cid:1)
ad([M,M ]) = [ad(M),ad(M )] = ad(M) ad(M ) ad(M ) ad(M) .
(cid:48) (cid:48) (cid:48) (cid:48)
◦ − ◦
The Adjoint action of the linear group GL(d) on (d) is by conju-
M
gation: GL(d) g Ad(g) is a morphism of groups, defined by:
(cid:51) (cid:55)→
1
Ad(g)(M) := gMg , for any g GL(d), M (d).
−
∈ ∈ M
A simple relation between the ad and Ad actions is as follows: for any
g GL(d),M (d), we have:
∈ ∈ M
(cid:0) (cid:1)
1
ad Ad(g)(M) = Ad(g) ad(M) Ad(g) . (I.1)
−
◦ ◦
Indeed, for any M(cid:48) we have:
∈ G
ad(cid:2)Ad(g)(M)(cid:3)(M(cid:48)) = g[M,g−1M(cid:48)g]g−1 = Ad(g) ad(M) Ad(g)−1(M(cid:48)).
◦ ◦
Using the exponential map (whose definition is recalled in the following
section I.1.2), we have moreover the following.
Lemma I.1.1.1 The differential of Ad exp at the unit 1 is ad. And:
◦
Ad(exp(A)) = exp[ad(A)] on (d), for any A (d). (I.2)
M ∈ M
R
Proof Consider the analytical map t Φ(t) := Ad(exp(tA)) from
(cid:55)→
into the space of endomorphisms on (d). It satisfies for any real t:
M
d d d
o o
Φ(t) = Ad(exp((s + t)A)) = Ad(exp(sA)exp(tA))
dt ds ds
d
o
= Φ(s) Φ(t) = ad(A) Φ(t).
ds ◦ ◦