**Mathematical Surveys and Monographs**

Volume: 114;
2005;
242 pp;
Softcover

MSC: Primary 30; 31; 60; 81; 82;

Print ISBN: 978-0-8218-4624-7

Product Code: SURV/114.S

List Price: $66.00

Individual Member Price: $52.80

**Electronic ISBN: 978-1-4704-1341-5
Product Code: SURV/114.S.E**

List Price: $66.00

Individual Member Price: $52.80

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#### Supplemental Materials

# Conformally Invariant Processes in the Plane

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*Gregory F. Lawler*

Theoretical physicists have predicted that the scaling limits of many two-dimensional lattice models in statistical physics are in some sense conformally invariant. This belief has allowed physicists to predict many quantities for these critical systems. The nature of these scaling limits has recently been described precisely by using one well-known tool, Brownian motion, and a new construction, the Schramm-Loewner evolution (SLE).

This book is an introduction to the conformally invariant processes that appear as scaling limits. The following topics are covered: stochastic integration; complex Brownian motion and measures derived from Brownian motion; conformal mappings and univalent functions; the Loewner differential equation and Loewner chains; the Schramm-Loewner evolution (SLE), which is a Loewner chain with a Brownian motion input; and applications to intersection exponents for Brownian motion. The prerequisites are first-year graduate courses in real analysis, complex analysis, and probability. The book is suitable for graduate students and research mathematicians interested in random processes and their applications in theoretical physics.

#### Table of Contents

# Table of Contents

## Conformally Invariant Processes in the Plane

- Contents v6 free
- Preface ix10 free
- Some discrete processes 114 free
- Chapter 1. Stochastic calculus 1124
- 1.1. Definition 1124
- 1.2. Integration with respect to Brownian motion 1225
- 1.3. Itô's formula 1730
- 1.4. Several Brownian motions 1831
- 1.5. Integration with respect to semimartingales 1932
- 1.6. Itô's formula for semimartingales 2033
- 1.7. Time changes of martingales 2235
- 1.8. Examples 2235
- 1.9. Girsanov's transformation 2336
- 1.10. Bessel processes 2538
- 1.11. Diffusions on an interval 3043
- 1.12. A Feynman-Kac formula 3952
- 1.13. Modulus of continuity 3952

- Chapter 2. Complex Brownian motion 4356
- Chapter 3. Conformal mappings 5770
- Chapter 4. Loewner differential equation 91104
- Chapter 5. Brownian measures on paths 119132
- Chapter 6. Schramm-Loewner evolution 147160
- 6.1. Chordal SLE 147160
- 6.2. Phases 150163
- 6.3. The locality property for K = 6 152165
- 6.4. The restriction property for K = 8/3 153166
- 6.5. Radial SLE 156169
- 6.6. Whole-plane SLE[sub(K)] 162175
- 6.7. Cardy's formula 163176
- 6.8. SLE[sub(6)] in an equilateral triangle 167180
- 6.9. Derivative estimates 169182
- 6.10. Crossing exponent for SLE[sub(6)] 171184
- 6.11. Derivative estimates, radial case 174187

- Chapter 7. More results about SLE 177190
- Chapter 8. Brownian intersection exponent 187200
- Chapter 9. Restriction measures 205218
- Appendix A. Hausdorff dimension 217230
- Appendix B. Hypergeometric functions 229242
- Appendix C. Reflecting Brownian motion 233246
- Bibliography 237250
- Index 240253
- Index of symbols 242255 free

#### Readership

Graduate students and research mathematicians interested in random processes and their applications in theoretical physics.

#### Reviews

This nice book celebrates the fruitful marriage of Brownian motion and complex analysis.

-- Zentralblatt MATH

This book gives a nice and systematic introduction to the contiuous time conformally invariant processes in the plane, assuming only knowledge of first year graduate real analysis, complex analysis and probability theory … This books is very well written, and can also be used as a graudate textbook for a topic course on SLE.

-- Mathematical Reviews