Classical potential theory and its probabilistic counterpart 3540412069, 9783540412069

Table of contents :
Contents......Page 8
Introduction......Page 24
Notation and Conventions......Page 28
Part 1 Classical and Parabolic Potential Theory ......Page 30
1. The Context of Green's Identity ......Page 32
3. Harmonic Functions ......Page 33
4. Maximum-Minimum Theorem for Harmonic Functions ......Page 34
5. The Fundamental Kernel for R" and Its Potentials ......Page 35
6. Gauss Integral Theorem ......Page 36
7. The Smoothness of Potentials; The Poisson Equation ......Page 37
8. Harmonic Measure and the Riesz Decomposition ......Page 40
1. The Green Function of a Ball; The Poisson Integral ......Page 43
2. Harnack's Inequality ......Page 45
3. Convergence of Directed Sets of Harmonic Functions ......Page 46
4. Harmonic, Subharmonic, and Superharmonic Functions ......Page 47
6. Application of the Operation tao_B ......Page 49
7. Characterization of Superharmonic Functions in Terms of Harmonic Functions ......Page 51
9. Application of Jensen's Inequality ......Page 52
10. Superharmonic Functions on an Annulus ......Page 53
11. Examples ......Page 54
12. The Kelvin Transformation (N>2) ......Page 55
14. The L' (? ) and D(? _) Classes of Harmonic Functions on a Ball B: The Riesz Herglotz Theorem ......Page 56
15 The Fatou Boundary Limit Theorem ......Page 60
16. Minimal Harmonic Functions ......Page 62
1. Least Superharmonic Majorant (LM) and Greatest Subharmonic Minorant (GM) ......Page 64
2. Generalization of Theorem I ......Page 65
3 Fundamental Convergence Theorem (Preliminary Version) ......Page 66
4. The Reduction Operation ......Page 67
5. Reduction Properties ......Page 70
6. A Smallness Property of Reductions on Compact Sets ......Page 71
7 The Natural (Pointwise) Order Decomposition for Positive Superharmonic Functions ......Page 72
1. Special Open Sets, and Potentials on Them ......Page 74
2. Examples ......Page 76
3. A Fundamental Smallness Property of Potentials ......Page 77
5. Smoothing of a Potential ......Page 78
6. Uniqueness of the Measure Determining a Potential ......Page 79
7. Riesz Measure Associated with a Superharmonic Function ......Page 80
8. Riesz Decomposition Theorem ......Page 81
9. Counterpart for Superharmonic Functions on R' of the Riesz Decomposition ......Page 82
10. An Approximation Theorem ......Page 84
1. Definition ......Page 86
2. Superharmonic Functions Associated with a Polar Set ......Page 87
4. Properties of Polar Sets ......Page 88
5. Extension of a Superharmonic Function ......Page 89
7. Superharmonic Function Minimum Theorem (Extension of Theorem 11.5) ......Page 92
8. Evans- Vasilesco Theorem ......Page 93
9. Approximation of a Potential by Continuous Potentials ......Page 95
10 The Domination Principle ......Page 96
11 The Infinity Set of a Potential and the Riesz Measure ......Page 97
1. The Fundamental Convergence Theorem ......Page 99
2. Inner Polar versus Polar Sets ......Page 100
3. Properties of the Reduction Operation ......Page 103
4. Proofs of the Reduction Properties ......Page 106
5. Reductions and Capacities ......Page 113
1. Definition of the Green Function G_D ......Page 114
2. Extremal Property of G_D ......Page 116
3. Boundedness Properties of G_D ......Page 117
4. Further Properties of G_D ......Page 119
5. The Potential G_Dnu of a Measure nu ......Page 121
7. The Existence of G_D versus the Greenian Character of D ......Page 123
9. Approximation Lemma ......Page 124
10. The Function G_D(.,xi)_{|D-xi|}, as a Minimal Harmonic Function ......Page 125
1. Relative Harmonic, Superharmonic, and Subharmonic Functions ......Page 127
2. The PWB Method ......Page 128
3. Examples ......Page 133
4. Continuous Boundary Functions on the Euclidean Boundary (h =1) ......Page 135
5. h-Harmonic Measure Null Sets ......Page 137
6. Properties of PWB^h Solutions ......Page 139
7. Proofs for Section 6 ......Page 140
8. h-Harmonic Measure ......Page 143
9. h-Resolutive Boundaries ......Page 147
10. Relations between Reductions and Dirichlet Solutions ......Page 151
11. Generalization of the Operator rB and Application to GM" ......Page 152
12. Barriers ......Page 153
13. h-Barriers and Boundary Point h-Regularity ......Page 155
14. Barriers and Euclidean Boundary Point Regularity ......Page 156
15. The Geometrical Significance of Regularity (Euclidean Boundary, h = 1) ......Page 157
16. Continuation of Section 13 ......Page 159
17. h-Harmonic Measure nu_D^h as a Function of D ......Page 160
18. The Extension G_D^= of G_D and the Harmonic Average G8 011 9) When D c B ......Page 161
19. Modification of Section 18 for D = 682 ......Page 165
20. Interpretation of 0D as a Green Function with Pole oo (N = 2) ......Page 168
21. Variant of the OperatortB ......Page 169
2. LMD u for an h-Subharmonic Function a ......Page 170
3. The Class D(Akl)-) ......Page 171
4. The Class L"(?-)(P ? I) ......Page 173
5. The Lattices (Si, 5) and (S`, E) ......Page 174
6. The Vector Lattice (S.