Contents
Preface ix
Some discrete processes 1
0.1. Simple random walk 1
0.2. Loop-erased random walk 3
0.3. Self-avoiding walk 5
0.4. Infinitely growing self-avoiding walk 7
0.5. Percolation exploration process 7
Chapter 1. Stochastic calculus 11
1.1. Definition 11
1.2. Integration with respect to Brownian motion 12
1.3. Ito's formula 17
1.4. Several Brownian motions 18
1.5. Integration with respect to semimartingales 19
1.6. Ito's formula for semimartingales 20
1.7. Time changes of martingales 22
1.8. Examples 22
1.9. Girsanov's transformation 23
1.10. Bessel processes 25
1.11. Diffusions on an interval 30
1.12. A Feynman-Kac formula 39
1.13. Modulus of continuity 39
Chapter 2. Complex Brownian motion 43
2.1. Review of complex analysis 43
2.2. Conformal invariance of Brownian motion 45
2.3. Harmonic functions 46
2.4. Green's function 52
Chapter 3. Conformal mappings 57
3.1. Simply connected domains 57
3.2. Univalent functions 60
3.3. Capacity 66
3.4. Half-plane capacity 69
3.5. Transformations on D 76
3.6. Caratheodory convergence 78
3.7. Extremal distance 80
3.8. Beurling estimate and applications 84
3.9. Conformal annuli 88
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