Stochastic analysis on manifolds / Elton P. Hsu.

Format:

Book

Author/Creator:

Hsu, Elton P., 1959-

Series:

Graduate studies in mathematics 1065-7339 ; v. 38.

Graduate studies in mathematics, 1065-7339 ; v. 38

Language:

English

Subjects (All):

Stochastic differential equations.

Diffusion processes.

Geometry, Differential.

Physical Description:

xiv, 281 pages ; 26 cm.

Place of Publication:

Providence, R.I. : American Mathematical Society, [2002]

Contents:

Chapter 1. Stochastic Differential Equations and Diffusions 5

1.1. SDE on euclidean space 6

1.2. SDE on manifolds 19

1.3. Diffusion processes 24

Chapter 2. Basic Stochastic Differential Geometry 35

2.1. Frame bundle and connection 36

2.2. Tensor fields 41

2.3. Horizontal lift and stochastic development 44

2.4. Stochastic line integrals 51

2.5. Martingales on manifolds 55

2.6. Martingales on submanifolds 60

Chapter 3. Brownian Motion on Manifolds 71

3.1. Laplace-Beltrami operator 72

3.2. Brownian motion on manifolds 78

3.3. Examples of Brownian motion 82

3.4. Distance function 88

3.5. Radial process 92

3.6. An exit time estimate 98

Chapter 4. Brownian Motion and Heat Semigroup 101

4.1. Heat kernel as transition density function 102

4.2. Stochastic completeness 107

4.3. C[subscript 0]-property of the heat semigroup 112

4.4. Recurrence and transience 117

4.5. Comparison of heat kernels 127

Chapter 5. Short-time Asymptotics 129

5.1. Short-time asymptotics: near points 130

5.2. Varadhan's asymptotic relation 133

5.3. Short-time asymptotics: distant points 137

5.4. Brownian bridge 142

5.5. Derivatives of the logarithmic heat kernel 147

Chapter 6. Further Applications 157

6.1. Dirichlet problem at infinity 158

6.2. Constant upper bound 166

6.3. Vanishing upper bound 168

6.4. Radially symmetric manifolds 172

6.5. Coupling of Brownian motion 179

6.6. Coupling and index form 182

6.7. Eigenvalue estimates 186

Chapter 7. Brownian Motion and Index Theorems 191

7.1. Weitzenbock formula 192

7.2. Heat equation on differential forms 198

7.3. Gauss-Bonnet-Chern formula 201

7.4. Clifford algebra and spin group 208

7.5. Spin bundle and the Dirac operator 213

7.6. Atiyah-Singer index theorem 217

7.7. Brownian holonomy 222

Chapter 8. Analysis on Path Spaces 229

8.1. Quasi-invariance of the Wiener measure 230

8.2. Flat path space 234

8.3. Gradient formulas 246

8.4. Integration by parts in path space 250

8.5. Martingale representation theorem 256

8.6. Logarithmic Sobolev inequality and hypercontractivity 258

8.7. Logarithmic Sobolev inequality on path space 263.

Notes:

Includes bibliographical references (pages 275-278) and index.

ISBN:

0821808028

OCLC:

47254307