The theory of Lévy processes in Lie groups is not merely an extension of the theory of Lévy processes in Euclidean spaces. Because of the unique structures possessed by non-commutative Lie groups, these processes exhibit certain interesting limiting properties which are not present for their counterparts in Euclidean spaces. This work provides an introduction to Lévy processes in general Lie groups, the limiting properties of Lévy processes in semi-simple Lie groups of non-compact type and the dynamical behavior of such processes as stochastic flows on certain homogeneous spaces.
Author(s): Ming Liao
Series: Cambridge tracts in mathematics 162
Publisher: Cambridge University Press
Year: 2004
Language: English
Pages: 278
City: Cambridge; New York
Cover......Page 1
Half-title......Page 3
Title......Page 5
Copyright......Page 6
Contents......Page 7
Preface......Page 9
List of Symbols......Page 11
Introduction......Page 13
1.1. Lévy Processes......Page 18
1.2. Generators of Lévy Processes......Page 22
1.3. Lévy Measure......Page 26
1.4. Stochastic Integral Equations......Page 31
1.5. Lévy Processes in GL(d, R)......Page 39
2.1. One-Point Motions......Page 44
2.2. Invariant Markov Processes in Homogeneous Spaces......Page 48
2.3. Riemannian Brownian Motions......Page 57
3.1. The Generator of a Lévy Process......Page 64
3.2. Existence and Uniqueness of the Solution to a Stochastic Integral Equation......Page 73
3.3. The Stochastic Integral Equation of a Lévy Process......Page 82
3.4. Generator of an Invariant Markov Process......Page 89
4.1. Fourier Analysis on Compact Lie Groups......Page 93
4.2. Lévy Processes in Compact Lie Groups......Page 96
4.3. Lévy Processes Invariant under the Inverse Map......Page 103
4.4. Conjugate Invariant Lévy Processes......Page 107
4.5. An Example......Page 112
5.1. Basic Properties of Semi-simple Lie Groups......Page 115
5.2. Roots and the Weyl Group......Page 117
5.3. Three Decompositions......Page 126
Special Linear Groups......Page 131
General Linear Groups......Page 135
Lorentz Groups......Page 136
6.1. Contracting Properties......Page 139
6.2. Limiting Properties: A Special Case......Page 144
6.3. A Continuous Lévy Process in GL(d, R)+......Page 150
6.4. Invariant Measures and Irreducibility......Page 152
6.5. Limiting Properties of Lévy Processes......Page 162
6.6. Some Sufficient Conditions......Page 172
7.1. Components under the Iwasawa Decomposition......Page 183
7.2. Rate of Convergence of the Abelian Component......Page 190
7.3. Haar Measure as Stationary Measure......Page 195
7.4. Rate of Convergence of the Nilpotent Component......Page 202
7.5. Right Lévy Processes......Page 208
8.1. Introduction to Stochastic Flows......Page 212
8.2. Lévy Processes as Dynamical Systems......Page 217
8.3. Lyapunov Exponents and Stable Manifolds......Page 222
8.4. Stationary Points and the Clustering Behavior......Page 231
8.5. SL(d, R)-flows......Page 235
8.6. SO(1, d)+-flow......Page 245
8.7. Stochastic Flows on S3......Page 249
A.1. Lie Groups......Page 256
A.2. Action of Lie Groups and Homogeneous Spaces......Page 259
B.1. Stochastic Processes......Page 261
B.2. Stochastic Integrals......Page 264
B.3. Poisson Random Measures......Page 268
Bibliography......Page 273
Index......Page 277
