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Gmndlehren der mathematischen Wissenschaften 262 A Series of Comprehensive Studies in Mathematics
Editors
M. Artin S. S. Chern J. M. Frohlich A. Grothendieck E. Heinz H. Hironaka F. Hirzebruch L. Hormander S. Mac Lane W. Magnus C. C. Moore J. K. Moser M. Nagata W. Schmidt D. S. Scott J. Tits B. L. van der Waerden M. Waldschmidt S. Watanabe
Managing Editors
M. Berger B. Eckmann S. R. S. Varadhan
Grundlehren der mathematischen Wissenschaften
A Series of Comprehensive Studies in Mathematics
A Selection
180. Landkof: Foundations of Modern Potential Theory 181. Lions/Magenes: Non-Homogeneous Boundary Value Problems and Applications I 182. Lions/Magenes: Non-Homogeneous Boundary Value Problems and Applications II
. 183. Lions/Magenes: Non-Homogeneous Boundary Value Problems and Applications III 184. Rosenblatt: Markov Processes, Structure and Asymptotic Behavior 185. Rubinowicz: Sommerfeldsche Polynommethode 186. Handbook for Automatic Computation. Vol. 2. Wilkinson/Reinsch: Linear Algebra 187. Siegel/Moser: Lectures on Celestial Mechanics 188. Warner: Harmonic Analysis on Semi-Simple Lie Groups I 189. Warner: Harmonic Analysis on Semi-Simple Lie Groups II 190. Faith: Algebra: Rings, Modules, and Categories I 191. Faith: Algebra II, Ring Theory 192. Mallcev: Algebraic Systems 193. P6Iya/Szego: Problems and Theorems in Analysis I 194. Igusa: Theta Functions 195. Berberian: Baer*-Rings 196. Athreya/Ney: Branching Processes 197. Benz: Vorlesungen tiber Geometric der Algebren 198. Gaal: Linear Analysis and Representation Theory 199. Nitsche: Voriesungen tiber Minimalfliichen 200. Dold: Lectures on Algebraic Topology 201. Beck: Continuous Flows in the Plane 202. Schmetterer: Introduction to Mathematical Statistics 203. Schoeneberg: Elliptic Modular Functions 204. Popov: Hyperstability of Control Systems 205. Nikollskii: Approximation of Functions of Several Variables and Imbedding Theorems 206. Andre: Homologie des Algebres Commutatives 207. Donoghue: Monotone Matrix Functions and Analytic Continuation 208. Lacey: The Isometric Theory of Classical Banach Spaces 209. Ringel: Map Color Theorem 210. Gihman/Skorohod: The Theory of Stochastic Processes I 211. Comfort!Negrepontis: The Theory of Ultrafilters 212. Switzer: Algebraic Topology-Homotopy and Homology 213. Shafarevich: Basic Algebraic Geometry 214. van der Waerden: Group Theory and Quantum Mechanics 215. Schaefer: Banach Lattices and Positive Operators 216. P6Iya/Szego: Problems and Theorems in Analysis II 217. Stenstrom: Rings of Quotients 218. Gihman/Skorohod: The Theory of Stochastic Process II 219. Duvant/Lions: Inequalities in Mechanics and Physics 220. Kirillov: Elements of the Theory of Representations 221. Mumford: Algebraic Geometry I: Complex Projective Varieties 222. Lang: Introduction to Modular Forms 223. Bergh/Lofstrom: Interpolation Spaces. An Introduction 224. Gilbarg/Trudinger: Elliptic Partial Differential Equations of Second Order
Continued after Index
J. L. Doob
Classical Potential Theory and Its Probabilistic Counterpart
Springer-Verlag New York Berlin Heidelberg Tokyo
J. L. Doob Department of Mathematics University of Illinois Urbana, IL 61801 U.S.A.
AMS Subject Classificaticns: 31-XX, 60J45
Library of Congress Cataloging in Publication Data Doob, Joseph L.
Classical potential theory and its probabilistic counterpart. (Grundlehren der mathematischen Wissenschaften; 262) Bibliography: p. l. Potential, Theory of. 2. Harmonic functions. 3. Martingales
(Mathematics) I. Title. II. Series. QA404.7.D66 1983 515.7 83-12446
© 1984 by Springer-Verlag New York Inc.
Softcover reprint of the hardcover 1 st edition 1984
All rights reserved. No part of this book may be translated or reproduced in any form without written consent from Springer-Verlag, 175 Fifth Avenue, New York, New York 10010, U.S.A.
Typeset by Asco Trade Typesetting Ltd., Hong Kong.
9 8 7 6 543 2 1
ISBN-13: 978-1-4612-9738-3 DOl: 10.1007/978-1-4612-5208-5
e-ISBN-13: 978-1-4612-5208-5
Contents
Introduction Notation and Conventions
Part 1 Classical and Parabolic Potential Theory
Chapter I Introduction to the Mathematical Background of Classical Potential
XXI
XXV
Theory .... " ..... " ................................. " . . . .. 3 1. The Context of Green's Identity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2. Function Averages ................... . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 3. Harmonic Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 4. Maximum-Minimum Theorem for Harmonic Functions ............... 5 5. The Fundamental Kernel for ~N and Its Potentials. . . . . . . . . . . . . . . . . . . . 6 6. Gauss Integral Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 7. The Smoothness of Potentials; The Poisson Equation. . . . . . . . . . . . . . . . . . 8 8. Harmonic Measure and the Riesz Decomposition. . . . . . . . . . . . . . . . . . . . . 11
Chapter II Basic Properties of Harmonic, Subharmonic, and Superharmonic Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1. The Green Function of a Ball; The Poisson Integral. . . . . . . . . . . . . . . . . . . 14 2. Harnack's Inequality ............................................. 16 3. Convergence of Directed Sets of Harmonic Functions ............... . . 17 4. Harmonic, Subharmonic, and Superharmonic Functions. . . . . . . . . . . . . . . 18 5. Minimum Theorem for Superharmonic Functions. . . . . . . . . . . . . . . . . . . . . 20 6. Application of the Operation 'B .................................... 20 7. Characterization of Superharmonic Functions in Terms of Harmonic
Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 8. Differentiable Superharmonic Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 9. Application of Jensen's Inequality. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
10. Superharmonic Functions on an Annulus. . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 11. Examples....................................................... 25 12. The Kelvin Transformation (N ~ 2). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 26
VI Contents
13. Greenian Sets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 14. The L1(IlB_) and D(IlB-) Classes of Harmonic Functions on a Ball B; The
Riesz-Herglotz Theorem .......... ; . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 15. The Fatou Boundary Limit Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 16. Minimal Harmonic Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
Chapter III Infima of Families of Superharmonic Functions .. . . . . . . . . . . . . . . . . 35
1. Least Superharmonic Majorant (LM) and Greatest Subharmonic Minorant (GM) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2. Generalization of Theorem 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 3. Fundamental Convergence Theorem (Preliminary Version) . . . . . . . . . . . . . 37 4. The Reduction Operation ......................................... 38 5. Reduction Properties ............................................. 41 6. A Smallness Property of Reductions on Compact Sets . . . . . . . . . . . . . . . . . 42 7. The Natural (Pointwise) Order Decomposition for Positive Superharmonic
Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
Chapter IV Potentials on Special Open Sets ................................ 45
1. Special Open Sets, and Potentials on Them. . . . . . . . . . . . . . . . . . . . . . . . . . . 45 2. Examples....................................................... 47 3. A Fundamental Smallness Property of Potentials ..................... 48 4. Increasing Sequences of Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 5. Smoothing of a Potential. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 6. Uniqueness of the Measure Determining a Potential................... 50 7. Riesz Measure Associated with a Superharmonic Function. . . . . . . . . . . . . 51 8. Riesz Decomposition Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 9. Counterpart for Superharmonic Functions on [R2 of the Riesz
Decomposition .................................................. 53 10. An Approximation Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
Chapter V Polar Sets and Their Applications .............................. 57
1. Definition....................................................... 57 2. Superharmonic Functions Associated with a Polar Set . . . . . . . . . . . . . . . . . 58 3. Countable Unions of Polar Sets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 4. Properties of Polar Sets ........................................... 59 5. Extension of a Superharmonic Function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 6. Greenian Sets in [R2 as the Complements of Nonpolar Sets . . . . . . . . . . . . . 63 7. Superharmonic Function Minimum Theorem (Extension of
Theorem 11.5). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 8. Evans-Vasilesco Theorem......................................... 64 9. Approximation of a Potential by Continuous Potentials. . . . . . . . . . . . . . . . 66
10. The Domination Principle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 11. The Infinity Set of a Potential and the Riesz Measure. . . . . . . . . . . . . . . . . . 68
Contents VB
Chapter VI The Fundamental Convergence Theorem and the Reduction Operation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
1. The Fundamental Convergence Theorem ............. . . . . . . . . . . . . . . . 70 2. Inner Polar versus Polar Sets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 3. Properties of the Reduction Operation .............................. 74 4. Proofs of the Reduction Properties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 5. Reductions and Capacities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
Chapter VII Green Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
1. Definition of the Green Function GD . . • . • • . • • • . • . • . . . . . • • • • • • • • • . • . • 85 2. Extremal Property of GD .•.•.•...••••.••.•••.•••.•.•••••.•••.•.•.• 87 3. Boundedness Properties of GD • • . • . • • • • • • • • • • • • • • • • • . • • • • • . • . • • • . . • • 88 4. Further Properties of GD •••••••••••••••••••••••••••••••••••••••••• 90 5. The Potential GDJ.1 of a Measure J.1 ...•••..•..•.••••••....•.•.••••.•. 92 6. Increasing Sequences of Open Sets and the Corresponding Green Function
Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 7. The Existence of GD versus the Greenian Character of D . . . . . . . . . . . . . . . 94 8. From Special to Greenian Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 9. Approximation Lemma............................ ....... ........ 95
10. The Function GD(" OID-{(} as a Minimal Harmonic Function. . . . . . . . . . . . 96
Chapter VIII The Dirichlet Problem for Relative Harmonic Functions. . . . . . . . . . . 98
1. Relative Harmonic, Superharmonic, and Subharmonic Functions . . . . . . . 98 2. The PWB Method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 3. Examples....................................................... 104 4. Continuous Boundary Functions on the Euclidean Boundary (h == 1) .... 106 5. h-Harmonic Measure Null Sets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 6. Properties of PWBh Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 7. Proofs for Section 6 .............................................. III 8. h-Harmonic Measure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 9. h-ResolutiveBoundaries ........................................... 118
10. Relations between Reductions and Dirichlet Solutions. . . . . . . . . . . . . . . . . 122 11. Generalization of the Operator "t"~ and Application to GMh . . . . . . . . . . . . . 123 12. Barriers......................................................... 124 13. h-Barriers and Boundary Point h-Regularity. . . . . . . . . . . . . . . . . . . . . . . . . . 126 14. Barriers and Euclidean Boundary Point Regularity. . . . . . . . . . . . . . . . . . . . 127 15. The Geometrical Significance of Regularity (Euclidean Boundary, h == I). 128 16. Continuation of Section 13 ........................................ 130 17. h-Harmonic Measure J.1t as a Function of D . . . . . . . . . . . . . . . . . . . . . . . . . . 131 18. The Extension G;; of GD and the Harmonic Average J.1D(~' G;('1, .)) When
DeB.......................................................... 132 19. Modification of Section 18 for D = [R2 •.......•....•............•.•. 136 20. Interpretation of ¢D as a Green Function with Pole 00 (N = 2) . . . . . . . . . . 139 21. Variant of the Operator"t"B......................................... 140
Vlll Contents
Chapter IX Lattices and Related Classes of Functions ....................... 141
1. Introduction..................................................... 141 2. LM~u for an h-Subharmonic Function u ............................ 141 3. The Class D(J.l~_) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 4. The Class LP(J.l~_)(p ~ 1). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 5. The Lattices (S±, :s:) and (S+, 6) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 6. The Vector Lattice (S,:5) ......................................... 146 7. The Vector Lattice Sm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 8. The Vector Lattice Sp . . . .. . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 9. The Vector Lattice Sqb .............•. . • • . . . . . . . . . • . • . • . . . . • • • . . . • . 150
10. The Vector Lattice Ss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 11. A Refinement of the Riesz Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 12. Lattices of h-Harmonic Functions on a Ball. . . . . . . . . . . . . . . . . . . . . . . . . . 152
Chapter X The Sweeping Operation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
1. Sweeping Context and Terminology. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 2. Relation between Harmonic Measure and the Sweeping Kernel. . . . . . . . . 157 3. Sweeping Symmetry Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . 158 4. Kernel Property of bt. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 5. Swept Measures and Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 6. Some Properties of bt. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 7. Poles of a Positive Harmonic Function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 8. Relative Harmonic Measure on a Polar Set 164
Chapter XI The Fine Topology. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
1. Definitions and Basic Properties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 2. A Thinness Criterion ....................................... . . . . . . 168 3. Conditions That ~ E A f . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 4. An Internal Limit Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 5. Extension of the Fine Topology to IRN u {oo } . . . . . . . . . . . . . . . . . . . . . . . . . 175 6. The Fine Topology Derived Set of a Subset of IRN . . . . . . . . . . . . . . . . . . . . . 177 7. Application to the Fundamental Convergence Theorem and to Reductions. 177 8. Fine Topology Limits and Euclidean Topology Limits. . . . . . . . . . . . . . . . . 178 9. Fine Topology Limits and Euclidean Topology Limits (Continued) . . . . . . 179
10. Identification of A f in Terms of a Special Function u# . . . . . . . . . . . . . . . . . 180 11. Quasi-Linde1i:if Property .......................................... 180 12. Regularity in Terms of the Fine Topology. . . . . . . . . . . . . . . . . . . . . . . . . . . 181 13. The Euclidean Boundary Set of Thinness of a Greenian Set. . . . . . . . . . . . . 182 14. The Support of a Swept Measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 15. Characterization of ~J.l~A .......................................... 183 16. A Special Reduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 17. The Fine Interior of a Set of Constancy of a Superharmonic Function ... 184 18. The Support of a Swept Measure (Continuation of Section 14) . . . . . . . . . . 185 19. Superharmonic Functions on Fine-Open Sets. . . . . . . . . . . . . . . . . . . . . . . . . 187 20. A Generalized Reduction.......................................... 187
Contents IX
21. Limits of Superharmonic Functions at Irregular Boundary Points of Their Domains. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
22. The Limit Harmonic Measure fJ1D •••••.••.••••••••••••••••••••••••• 191 23. Extension of the Domination Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194
Chapter XII The Martin Boundary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
1. Motivation...................................................... 195 2. The Martin Functions ............................................ 196 3. The Martin Space. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 4. Preliminary Representations of Positive Harmonic Functions and Their
Reductions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 5. Minimal Harmonic Functions and Their Poles ....................... 200 6. Extension of Lemma 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 7. The Set of Nonminimal Martin Boundary Points ..................... 202 8. Reductions on the Set of Minimal Martin Boundary Points ............ 203 9. The Martin Representation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204
10. Resolutivity of the Martin Boundary.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 11. Minimal Thinness at a Martin Boundary Point . . . . . . . . . . . . . . . . . . . . . . . 208 12. The Minimal-Fine Topology. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210 13. First Martin Boundary Counterpart of Theorem XI.4(c) and (d) ........ 213 14. Second Martin Boundary Counterpart of Theorem XI.4(c) . . . . . . . . . . . . . 213 15. Minimal-Fine Topology Limits and Martin Topology Limits at a Minimal
Martin Boundary Point. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215 16. Minimal-Fine Topology Limits and Martin Topology Limits at a Minimal
Martin Boundary Point (Continued) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216 17. Minimal-Fine Martin Boundary Limit Functions..................... 216 18. The Fine Boundary Function of a Potential. . . . . . . . . . . . . . . . . . . . . . . . . . 218 19. The Fatou Boundary Limit Theorem for the Martin Space. . . . . . . . . . . . . 219 20. Classical versus Minimal-Fine Topology Boundary Limit Theorems for
Relative Superharmonic Functions on a Ball in [RN • . . . . . . . . . . . • • . . . . . . 221 21. Nontangential and Minimal-Fine Limits at a Half-space Boundary. . . . . . 222 22. Normal Boundary Limits for a Half-space. . . . . . . . . . . . . . . . . . . . . . . . . . . 223 21. Boundary Limit Function (Minimal-Fine and Normal) of a Potential on a
Half-space ................................................ . . . . . . 225
Chapter XIII Classical Energy and Capacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226
1. Physical Context . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226 2. Measures and Their Energies ...................................... 227 3. Charges and Their Energies. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228 4. Inequalities between Potentials, and the Corresponding Energy
Inequalities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 5. The Function DHGDfl............................................ 230 6. Classical Evaluation of Energy; Hilbert Space Methods. . . . . . . . . . . . . . . . 231 7. The Energy Functional (Relative to an Arbitrary Greenian Subset D of
[RN). • . • . . . • • • • • • • • • • • . . • . . . • . • • • • . . • • • . • . . . . • • . • . . • . • . • . . • • • • • • • 233 8. Alternative Proofs of Theorem 7(b+) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235
x Contents
9. Sharpening of Lemma 4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237 10. The Classical Capacity Function ................................... 237 II. Inner and Outer Capacities (Notation of Section 10). . . . . . . . . . . . . . . . . . . 240 12. Extremal Property Characterizations of Equilibrium Potentials (Notation
of Section 10). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241 13. Expressions for C(A) ............................................. 243 14. The Gauss Minimum Problems and Their Relation to Reductions. . . . . . . 244 15. Dependence of C* on D. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247 16. Energy Relative to ~2 • • . • • • • . . . . . . • . . • • • • • . . . . . . . . . . • . • • • • • • • • • • • . 248 17. The Wiener Thinness Criterion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249 18. The Robin Constant and Equilibrium Measures Relative to ~2 (N = 2) . . 251
Chapter XIV One-Dimensional Potential Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256
1. Introduction ...................................... ' . . . . . . . . . . . . . . . 256 2. Harmonic, Superharmonic, and Subharmonic Functions. . . . . . . . . . . . . . . 256 3. Convergence Theorems. . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256 4. Smoothness Properties of Superharmonic and Subharmonic Functions. . . 257 5. The Dirichlet Problem (Euclidean Boundary). . . . . . . . . . . . . . . . . . . . . . . . . 257 6. Green Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258 7. Potentials of Measures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259 8. Identification of the Measure Defining a Potential ..................... 259 9. Riesz Decomposition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260
10. The Martin Boundary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261
Chapter XV Parabolic Potential Theory: Basic Facts . . . . . . . . . . . . . . . . . . . . . . . . . 262
1. Conventions..................................................... 262 2. The Parabolic and Coparabolic Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . 263 3. Coparabolic Polynomials. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264 4. The Parabolic Green Function of IRN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266 5. Maximum-Minimum Parabolic Function Theorem. . . . . . . . . . . . . . . . . . . . 267 6. Application of Green's Theorem ................................... 269 7. The Parabolic Green Function of a Smooth Domain; The Riesz Decom
position and Parabolic Measure (Formal Treatment) . . . . . . . . . . . . . . . . . . 270 8. The Green Function of an Interval. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272 9. Parabolic Measure for an Interval. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273
10. Parabolic Averages. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275 11. Harnack's Theorems in the Parabolic Context. . . . . . . . . . . . . . . . . . . . . . . . 276 12. Superparabolic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277 13. Superparabolic Function Minimum Theorem ........................ 279 14. The Operation iIi and the Defining Average Properties of Superparabolic
Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280 15. Superparabolic and Parabolic Functions on a Cylinder ................ 281 16. The Appell Transformation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282 17. Extensions of a Parabolic Function Defined on a Cylinder ............. 283
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Chapter XVI Subparabolic, Superparabolic, and Parabolic Functions on a Slab. .. 285
1. The Parabolic Poisson Integral for a Slab. . . . . . . . . . . . . . . . . . . . . . . . . . . . 285 2. A Generalized Superparabolic Function Inequality. . . . . . . . . . . . . . . . . . . . 287 3. A Criterion of a Subparabolic Function Supremum ... . . . . . . . . . . . . . . . . 288 4. A Boundary Limit Criterion for the Identically Vanishing of a Positive
Parabolic Function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288 5. A Condition that a Positive Parabolic Function Be Representable by a
Poisson Integral. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290 6. The Ll(tili_) and DCtili-) Classes of Parabolic Functions on a Slab...... 290 7. The Parabolic Boundary Limit Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292 8. Minimal Parabolic Functions on a Slab ............................. 293
Chapter XVII Parabolic Potential Theory (Continued) . . . . . . . . . . . . . . . . . . . . . . . . . 295
1. Greatest Minorants and Least Majorants . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295 2. The Parabolic Fundamental Convergence Theorem (Preliminary Version)
and the Reduction Operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295 3. The Parabolic Context Reduction Operations ........................ 296 4. The Parabolic Green Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298 5. Potentials....................................................... 300 6. The Smoothness of Potentials. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303 7. Riesz Decomposition Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305 8. Parabolic-Polar Sets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305 9. The Parabolic-Fine Topology. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308
10. Semipolar Sets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309 11. Preliminary List of Reduction Properties ............................ 310 12. A Criterion of Parabolic Thinness .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313 13. The Parabolic Fundamental Convergence Theorem. . . . . . . . . . . . . . . . . . . 314 14. Applications of the Fundamental Convergence Theorem to Reductions
and to Green Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316 15. Applications of the Fundamental Convergence Theorem to the Parabolic-
Fine Topology. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317 16. Parabolic-Reduction Properties .............................. . . . . . . 317 17. Proofs of the Reduction Properties in Section 16 . . . . . . . . . . . . . . . . . . . . . . 320 18. The Classical Context Green Function in Terms of the Parabolic Context
Green Function (N ~ 1). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 326 19. The Quasi-Linde1of Property. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328
Chapter XVIII The Parabolic Dirichlet Problem, Sweeping, and Exceptional Sets. . . 329
1. Relativization of the Parabolic Context; The PWB Method in this Context. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329
2. h-Parabolic Measure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332 3. Parabolic Barriers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333 4. Relations between the Classical Dirichlet Problem and the Parabolic
Context Dirichlet Problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334 5. Classical Reductions in the Parabolic Context. . . . . . . . . . . . . . . . . . . . . . . . 335
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6. Parabolic Regularity of Boundary Points ............................ 337 7. Parabolic Regularity in Terms of the Fine Topology. . . . . . . . . . . . . . . . . . . 341 8. Sweeping in the Parabolic Context. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341 9. The Extension 0; of aD and the Parabolic Average J.iD(~,a;(·,~)) when
iJ c iJ .......................................................... 343 10. Conditions that ~ E Api. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345 II. Parabolic- and Coparabolic-Polar Sets .............................. 347 12. Parabolic- and Coparabolic-Semipolar Sets . . . . . . . . . . . . . . . . . . . . . . . . . . 348 13. The Support of a Swept Measure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350 14. An Internal Limit Theorem; The Coparabolic-Fine Topology Smoothness
of Superparabolic Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351 15. Application to a Version of the Parabolic Context Fatou Boundary Limit
Theorem on a Slab . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357 16. The Parabolic Context Domination Principle. . . . . . . . . . . . . . . . . . . . . . . . . 358 17. Limits of Superparabolic Functions at Parabolic-Irregular Boundary
Points of Their Domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 358 18. Martin Flat Point Set Pairs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361 19. Lattices and Related Classes of Functions in the Parabolic Context. . . . . . 361
Chapter XIX The Martin Boundary in the Parabolic Context. . . . . . . . . . . . . . . . . . . 363
1. Introduction..................................................... 363 2. The Martin Functions of Martin Point Set and Measure Set Pairs. . . . . . . 364 3. The Martin Space iJM ............................................ 366 4. Preparatory Material for the Parabolic Context Martin Representation
Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367 5. Minimal Parabolic Functions and Their Poles . . . . . . . . . . . . . . . . . . . . . . . . 369 6. The Set of Nonminimal Martin Boundary Points ..................... 370 7. The Martin Representation in the Parabolic Context .................. 371 8. Martin Boundary of a Slab iJ = ~N X ]0,.5[ with 0 < .5 ~ + 00 ......... 371 9. Martin Boundaries for the Lower Half-space of ~N and for ~N • • • • . . • • • • 374
10. The Martin Boundary of iJ = ]0, + oo[ x ] - 00,.5[ ................... 375 II. PWBh Solutions on iJM ........................................... 377 12. The Minimal-Fine Topology in the Parabolic Context. . . . . . . . . . . . . . . . . 377 13. Boundary Counterpart of Theorem XVIII.l4(f) ... . . . . . . . . . . . . . . . . . . . 379 14. The Vanishing of Potentials on OM iJ ................................ 381 15. The Parabolic Context Fatou Boundary Limit Theorem on Martin Spaces 381
Part 2 Probabilistic Counterpart of Part 1
Chapter I Fundamental Concepts of Probability. . . . . . . . . . . . . . . . . . . . . . . . . . . 387
I. Adapted Families of Functions on Measurable Spaces. . . . . . . . . . . . . . . . . 387 2. Progressive Measurability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388 3. Random Variables. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 390
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4. Conditional Expectations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 391 5. Conditional Expectation Continuity Theorem. . . . . . . . . . . . . . . . . . . . . . . . 393 6. Fatou's Lemma for Conditional Expectations ... . . . . . . . . . . . . . . . . . . . . . 396 7. Dominated Convergence Theorem for Conditional Expectations. . . . . . . . 397 8. Stochastic Processes, "Evanescent," "Indistinguishable," "Standard Modi-
fication," "Nearly" . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 398 9. The Hitting of Sets and Progressive Measurability .................... 401
10. Canonical Processes and Finite-Dimensional Distributions . . . . . . . . . . . . . 402 11. Choice of the Basic Probability Space. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404 12. The Hitting of Sets by a Right Continuous Process. . . . . . . . . . . . . . . . . . . . 405 13. Measurability versus Progressive Measurability of Stochastic Processes .. 407 14. Predictable Families of Functions .................................. 410
Chapter II Optional Times and Associated Concepts. . . . . . . . . . . . . . . . . . . . . . . . 413
1. The Context of Optional Times.................................... 413 2. Optional Time Properties (Continuous Parameter Context). . . . . . . . . . . . . 415 3. Process Functions at Optional Times. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417 4. Hitting and Entry Times. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419 5. Application to Continuity Properties of Sample Functions ............. 421 6. Continuation of Section 5 ......................................... 423 7. Predictable Optional Times. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423 8. Section Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425 9. The Graph of a Predictable Time and the Entry Time of a Predictable
Set....................................................... ...... 426 10. Semipolar Subsets of IR + x Q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 427 11. The Classes D and LP of Stochastic Processes. . . . . . . . . . . . . . . . . . . . . . . . . 428 12. Decomposition of Optional Times; Accessible and Totally Inaccessible
Optional Times . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429
Chapter III Elements of Martingale Theory ................................ 432
1. Definitions...................................................... 432 2. Examples....................................................... 433 3. Elementary Properties (Arbitrary Simply Ordered Parameter Set) ....... 435 4. The Parameter Set in Martingale Theory ............................ 437 5. Convergence of Supermartingale Families ........................... 437 6. Optional Sampling Theorem (Bounded Optional Times) . . . . . . . . . . . . . . . 438 7. Optional Sampling Theorem for Right Closed Processes . . . . . . . . . . . . . . . 440 8. Optional Stopping. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 442 9. Maximal Inequalities ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 442
10. Conditional Maximal Inequalities .................................. 444 11. An U Inequality for Submartingale Suprema. . . . . . . . . . . . . . . . . . . . . . . . . 444 12. Crossings....................................................... 445 13. Forward Convergence in the L 1 Bounded Case. . . . . . . . . . . . . . . . . . . . . . . 450 14. Convergence of a Uniformly Integrable Martingale. . . . . . . . . . . . . . . . . . . 451 15. Forward Convergence of a Right Closable Supermartingale .... . . . . . . . . 453 16. Backward Convergence of a Martingale. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454
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17. Backward Convergence ofa Supermartingale......................... 455 18. The 't Operator .................................................. 455 19. The Natural Order Decomposition Theorem for Supermartingales ...... 457 20. The Operators LM and GM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 458 21. Supermartingale Potentials and the Riesz Decomposition. . . . . . . . . . . . . . 459 22. Potential Theory Reductions in a Discrete Parameter Probability
Context. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 459 23. Application to the Crossing Inequalities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 461
Chapter IV Basic Properties of Continuous Parameter Supermartingales. . . . . . . . 463
1. Continuity Properties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463 2. Optional Sampling of Uniformly Integrable Continuous Parameter
Martingales ..................................................... 468 3. Optional Sampling and Convergence of Continuous Parameter
Supermartingales. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 470 4. Increasing Sequences of Supermartingales ........................... 473 5. Probability Version of the Fundamental Convergence Theorem of Potential
Theory......................................................... 476 6. Quasi-Bounded Positive Supermartingales; Generation ofSupermartingale
Potentials by Increasing Processes .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 480 7. Natural versus Predictable Increasing Processes (/ = Z+ or [R+) . . . . . . . . . 483 8. Generation of Supermartingale Potentials by Increasing Processes in the
Discrete Parameter Case .......................................... 488 9. An Inequality for Predictable Increasing Processes. . . . . . . . . . . . . . . . . . . . 489
10. Generation of Supermartingale Potentials by Increasing Processes for Arbitrary Parameter Sets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 490
11. Generation of Supermartingale Potentials by Increasing Processes in the Continuous Parameter Case: The Meyer Decomposition. . . . . . . . . . . . . . . 493
12. Meyer Decomposition of a Submartingale . . . . . . . . . . . . . . . . . . . . . . . . . . . 495 13. Role of the Measure Associated with a Supermartingale;
The Supermartingale Domination Principle . . . . . . . . . . . . . . . . . . . . . . . . . . 496 14. The Operators 't, LM, and GM in the Continuous Parameter Context. . . . 500 15. Potential Theory on IR+ x Q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 501 16. The Fine Topology of [R+ x Q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 502 17. Potential Theory Reductions in a Continuous Parameter Probability
Context. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504 18. Reduction Properties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 505 19. Proofs of the Reduction Properties in Section 18 . . . . . . . . . . . . . . . . . . . . . . 509 20. Evaluation of Reductions ......................................... 513 21. The Energy of a Supermartingale Potential. . . . . . . . . . . . . . . . . . . . . . . . . . . 515 22. The Subtraction of a Supermartingale Discontinuity. . . . . . . . . . . . . . . . . . . 516 23. Supermartingale Decompositions and Discontinuities. . . . . . . . . . . . . . . . . 518
Chapter V Lattices and Related Classes of Stochastic Processes. . . . . . . . . . . . . . . 520
1. Conventions; The Essential Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 520 2. LM x(.) when {x(·), ff(.)} Is a Submartingale ........................ 521
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3. Uniformly Integrable Positive Submartingales . . . . . . . . . . . . . . . . . . . . . . . . 523 4. U Bounded Stochastic Processes (p ~ 1) ... . . . . . . . . . . . . . . . . . . . . . . . . . 524 5. The Lattices ('S±, :$;), ('S+, :$;), (S±, :$;), (S+,:$;) ..................... 525 6. The Vector Lattices ('S, :S) and (S,:S) .............................. 528 7. The Vector Lattices ('Sm,:S) and (Sm, :S) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 529 8. The Vector Lattices ('Sp,:S) and (Sp, :S). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 530 9. The Vector Lattices (,Sqb,:S) and (Sqb,:S) ........................... 531
10. The Vector Lattices ('S., s) and (S.,:S) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 532 11. The Orthogonal Decompositions 'Sm = 'Smqb + 'Sms and Sm = Smqb + Sms . 533 12. Local Martingales and Singular Supermartingale Potentials in (S, s) . . . . 534 13. Quasimartingales (Continuous Parameter Context) . . . . . . . . . . . . . . . . . . . . 535
Chapter VI Markov Processes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 539
1. The Markov Property. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 539 2. Choice of Filtration .............................................. 544 3. Integral Parameter Markov Processes with Stationary Transition Proba-
bilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 545 4. Application of Martingale Theory to Discrete Parameter Markov
Processes ....................................................... 547 5. Continuous Parameter Markov Processes with Stationary Transition
Probabilities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 550 6. Specialization to Right Continuous Processes ........................ 552 7. Continuous Parameter Markov Processes: Lifetimes and Trap Points. . . . 554 8. Right Continuity of Markov Process Filtrations; A Zero-One (0-1) Law. . 556 9. Strong Markov Property. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 557
10. Probabilistic Potential Theory; Excessive Functions. . . . . . . . . . . . . . . . . . . 560 II. Excessive Functions and Supermartingales . . . . . . . . . . . . . . . . . . . . . . . . . . . 564 12. Excessive Functions and the Hitting Times of Analytic Sets (Notation and
Hypotheses of Section 11) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 565 13. Conditioned Markov Processes..................................... 566 14. Tied Down Markov Processes...................................... 567 15. Killed Markov Processes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 568
Chapter VII Brownian Motion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 570
1. Processes with Independent Increments and State Space /RN .•.••••••••• 570 2. Brownian Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 572 3. Continuity of Brownian Paths. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 576 4. Brownian Motion Filtrations ...................................... 578 5. Elementary Properties of the Brownian Transition Density and Brownian
Motion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 581 6. The Zero-One Law for Brownian Motion. . . . . . . . . . . . . . . . . . . . . . . . . . . . 583 7. Tied Down Brownian Motion. . .. . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 586 8. Andre Reflection Principle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 587 9. Brownian Motion in an Open Set (N ~ I)............................ 589
10. Space-Time Brownian Motion in an Open Set. . . . . . . . . . . . . . . . . . . . . . . . 592 11. Brownian Motion in an Interval. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 594
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12. Probabilistic Evaluation of Parabolic Measure for an Interval .......... 595 13. Probabilistic Significance of the Heat Equation and Its Dual 596
Chapter VIII The Ito Integral. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 599 I. Notation........................................................ 599 2. The Size offo ................................................... 601 3. Properties of the Ito Integral. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 602 4. The Stochastic Integral for an Integrand Process in fo . . . . . . . . . . . . . . . . . 605 5. The Stochastic Integral for an Integrand Process in f. . . . . . . . . . . . . . . . . . 606 6. Proofs of the Properties in Section 3 ................................ 607 7. Extension to Vector-Valued and Complex-Valued Integrands........... 611 8. Martingales Relative to Brownian Motion Filtrations ................. 612 9. A Change of Variables ................. .'.......................... 615
10. The Role of Brownian Motion Increments........................... 618 11. (N = 1) Computation of the Ito Integral by Riemann-Stieltjes Sums. . . . . 620 12. Ito's Lemma. . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 621 13. The Composition of the Basic Functions of Potential Theory with Brownian
Motion......................................................... 625 14. The Composition of an Analytic Function with Brownian Motion. . . . . . . 626
Chapter IX Brownian Motion and Martingale Theory ....................... 627
1. Elementary Martingale Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 627 2. Coparabolic Polynomials and Martingale Theory. . . . . . . . . . . . . . . . . . . . . 630 3. Superharmonic and Harmonic Functions on [RN and Supermartingales and
Martingales ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 632 4. Hitting of an Fa Set .................................. ~ . . . . . . . . . . . . 635 5. The Hitting of a Set by Brownian Motion. . . . . . . . . . . . . . . . . . . . . . . . . . . . 636 6. Superharmonic Functions, Excessive for Brownian Motion. . . . . . . . . . . . . 637 7. Preliminary Treatment of the Composition of a Superharmonic Function
with Brownian Motion; A Probabilistic Fatou Boundary Limit Theorem. 641 8. Excessive and Invariant Functions for Brownian Motion. . . . . . . . . . . . . . . 645 9. Application to Hitting Probabilities and to Parabolicity of Transition
Densities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 647 10. (N = 2). The Hitting of Nonpolar Sets by Brownian Motion. . . . . . . . . . . . 648 11. Continuity of the Composition of a Function with Brownian Motion . . . . 649 12. Continuity of Superharmonic Functions on Brownian Motion. . . . . . . . . . 650 13. Preliminary Probabilistic Solution of the Classical Dirichlet Problem .... 651 14. Probabilistic Evaluation of Reductions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 653 15. Probabilistic Description of the Fine Topology. . . . . . . . . . . . . . . . . . . . . . . 656 16. oc-Excessive Functions for Brownian Motion and Their Composition with
Brownian Motions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 659 17. Brownian Motion Transition Functions as Green Functions; The Corre-
sponding Backward and Forward Parabolic Equations. . . . . . . . . . . . . . . . 661 18. Excessive Measures for Brownian Motion. . . . . . . . . . . . . . . . . . . . . . . . . . . 663 19. Nearly Borel Sets for Brownian Motion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 666 20. Brownian Motion into a Set from an Irregular Boundary Point . . . . . . . . . 666
Contents xvii
Chapter X Conditional Brownian Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 668
1. Definition....................................................... 668 2. h-Brownian Motion in Terms of Brownian Motion. . . . . . . . . . . . . . . . . . . . 671 3. Contexts for (2.1) ................................................ 676 4. Asymptotic Character of h-Brownian Paths at Their Lifetimes. . . . . . . . . . 677 5. h-Brownian Motion from an Infinity of h . . . . . . . . . . . . . . . . . . . . . . . . . . . . 680 6. Brownian Motion under Time Reversal ............................. 682 7. Preliminary Probabilistic Solution of the Dirichlet Problem for h-Harmonic
Functions; h-Brownian Motion Hitting Probabilities and the Corresponding Generalized Reductions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 684
8. Probabilistic Boundary Limit and Internal Limit Theorems for Ratios of Strictly Positive Superharmonic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 688
9. Conditional Brownian Motion in a Ball ............................. 691 10. Conditional Brownian Motion Last Hitting Distributions; The Capacitary
Distribution of a Set in Terms of a Last Hitting Distribution ........... 693 II. The Tail (J Algebra of a Conditional Brownian Motion . . . . . . . . . . . . . . . . 694 12. Conditional Space-Time Brownian Motion. . . . . . . . . . . . . . . . . . . . . . . . . . 699 13. [Space-Time] Brownian Motion in [~N] IRN with Parameter Set IR....... 700
Part 3
Chapter I Lattices in Classical Potential Theory and Martingale Theory. . . . . . . 705
I. Correspondence between Classical Potential Theory and Martingale Theory......................................................... 705
2. Relations between Decomposition Components of S in Potential Theory and Martingale Theory ..................................... . . . . . . 706
3. The Classes LP and D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 706 4. PWB-Related Conditions on h-Harmonic Functions and on Martingales. 707 5. Class D Property versus Quasi-Boundedness ......................... 708 6. A Condition for Quasi-Boundedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 709 7. Singularity of an Element ofS~ .................................... 710 8. The Singular Component of an Element of S+ . . . . . . . . . . . . . . . . . . . . . . . . 711 9. The Class Spqb . . . . . • . . • • . . . . . . . . • . . • . • • . • . . . . • . • . • • • • . . • . . . • • . • • . 712
10. The Class Sps • • . . • • • . • • . . . . • • • • • • • • • • • • • • • • • • . . • • . . . . • • . . • . • • • • • . 714 11. Lattice Theoretic Analysis of the Composition of an h-Superharmonic
Function with an h-Brownian Motion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 715 12. A Decomposition ofS~s (Potential Theory Context). . . . . . . . . . . . . . . . . . . 716 13. Continuation of Section II ........................................ 717
Chapter II Brownian Motion and the PWB Method.. . ... .... .. .. .. . .. .. . .. 719
1. Context of the Problem........................................... 719 2. Probabilistic Analysis of the PWB Method. . . . . . . . . . . . . . . . . . . . . . . . . . . 720
xviii Contents
3. PWBh Examples ................................................. 723 4. Tail (J Algebras in the PWBh Context. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 725
Chapter III Brownian Motion on the Martin Space. . . . . . . . . . . . . . . . . . . . . . . . . . 727
I. The Structure of Brownian Motion on the Martin Space . . . . . . . . . . . . . . . 727 2. Brownian Motions from Martin Boundary Points (Notation of Section I) 728 3. The Zero-One Law at a Minimal Martin Boundary Point and the
Probabilistic Formulation of the Minimal-Fine Topology (Notation of Section I) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 730
4. The Probabilistic Fatou Theorem on the Martin Space. . . . . . . . . . . . . . . . . 732 5. Probabilistic Approach to Theorem I.XI.4(c) and Its Boundary
Counterparts .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 733 6. Martin Representation of Harmonic Functions in the Parabolic Context. 735
Appendixes
Appendix I Analytic Sets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 741
1. Pavings and Algebras of Sets. ...................................... 741 2. Suslin Schemes .................................................. 741 3. Sets Analytic over a Product Paving ................................ 742 4. Analytic Extensions versus (J Algebra Extensions of Pavings. . . . . . . . . . . . 743 5. Projection Characterization d(qy) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 743 6. The Operation d(d) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 744 7. Projections of Sets in Product Pavings. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 744 8. Extension of a Measurability Concept to the Analytic Operation Context. 745 9. The G~ Sets of a Complete Metric Space. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 745
10. Polish Spaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 746 11. The Baire Null Space. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 746 12. Analytic Sets .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 747 13. Analytic Subsets of Polish Spaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 748
Appendix II Capacity Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 750
1. Choquet Capacities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 750 2. Sierpinski Lemma. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 750 3. Choquet Capacity Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 751 4. Lusin's Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 751 5. A Fundamental Example of a Choquet Capacity. . . . . . . . . . . . . . . . . . . . . . 752 6. Strongly Subadditive Set Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 752 7. Generation of a Choquet Capacity by a Positive Strongly Subadditive Set
Function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 753 8. Topological Precapacities ......................................... 755 9. Universally Measurable Sets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 756
Contents XIX
Appendix III Lattice Theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 758
1. Introduction..................................................... 758 2. Lattice Definitions ............................................... 758 3. Cones.......................................................... 758 4. The Specific Order Generated by a Cone ............................ 759 5. Vector Lattices .................................................. 760 6. Decomposition Property of a Vector Lattice ......................... 762 7. Orthogonality in a Vector Lattice. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 762 8. Bands in a Vector Lattice. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 762 9. Projections on Bands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 763
10. The Orthogonal Complement of a Set. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 764 II. The Band Generated by a Single Element. . . . . . . . . . . . . . . . . . . . . . . . . . . . 764 12. Order Convergence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 765 13. Order Convergence on a Linearly Ordered Set. . . . . . . . . . . . . . . . . . . . . . . . 766
Appendix IV Lattice Theoretic Concepts in Measure Theory . . . . . . . . . . . . . . . . . . . 767
I. Lattices of Set Algebras. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . 767 2. Measurable Spaces and Measurable Functions ....................... 767 3. Composition of Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 768 4. The Measure Lattice of a Measurable Space. . . . . . . . . . . . . . . . . . . . . . . . . . 769 5. The (J Finite Measure Lattice of a Measurable Space (Notation of Section 4) 771 6. The Hahn and Jordan Decompositions. . . . . . . .. . . . . . . . . . . . . . . . . . . . . . 772 7. The Vector Lattice.A" . . . . .. . . . . . . . .. . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 772 8. Absolute Continuity and Singularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 773 9. Lattices of Measurable Functions on a Measure Space. . . . . . . . . . . . . . . . . 774
10. Order Convergence of Families of Measurable Functions.............. 775 II. Measures on Polish Spaces ........................................ 777 12. Derivates of Measures .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 778
Appendix V Uniform Integrability 779
Appendix VI Kernels and Transition Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 781
I. Kernels......................................................... 781 2. Universally Measurable Extension of a Kernel. . . . . . . . . . . . . . . . . . . . . . . . 782 3. Transition Functions ............................................. 782
Appendix VII Integral Limit Theorems ...................................... 785
1. An Elementary Limit Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 785 2. Ratio Integral Limit Theorems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 786 3. A One-Dimensional Ratio Integral Limit Theorem. . . . . . . . . . . . . . . . . . . . 786 4. A Ratio Integral Limit Theorem Involving Convex Variational Derivates. 788
xx Contents
Appendix VIII Lower Semicontinuous Functions ............................... 791
1. The Lower Semicontinuous Smoothing of a Function. . . . . . . . . . . . . . . . . 791 2. Suprema of Families of Lower Semicontinuous Functions.............. 791 3. Choquet Topological Lemma. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 792
Historical Notes ............................................. 793 Part 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 793 Part 2............................................................... 806 Part 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 815 Appendixes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 816
Bibliography. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 819 Notation Index. . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 827 Index... . ... . . ... . . ...... . .. ... .. ... .. . . ... .. . .. .. . . .... . . .. 829
Introduction
Potential theory and certain aspects of probability theory are intimately related, perhaps most obviously in that the transition function determining a Markov process can be used to define the Green function of a potential theory. Thus it is possible to define and develop many potential theoretic concepts probabilistically, a procedure potential theorists observe withjaundiced eyes in view of the fact that now as in the past their subject provides the motivation for much of Markov process theory. However that may be it is clear that certain concepts in potential theory correspond closely to concepts in probability theory, specifically to concepts in martingale theory. For example, superharmonic functions correspond to supermartingales. More specifically: the Fatou type boundary limit theorems in potential theory correspond to supermartingale convergence theorems; the limit properties of monotone sequences of superharmonic functions correspond surprisingly closely to limit properties of monotone sequences of supermartingales; certain positive superharmonic functions [supermartingales] are called "potentials," have associated measures in their respective theories and are subject to domination principles (inequalities) involving the supports of those measures; in each theory there is a reduction operation whose properties are the same in the two theories and these reductions induce sweeping (balayage) of the measures associated with potentials, and so on.
The purpose of this book is to develop this correspondence between potential theory and probability theory by examining in detail classical potential theory, that is, the potential theory of Laplace's equation, together with the corresponding probability theory, that is, martingale theory. The joining link which makes this correspondence especially perspicuous is the Brownian motion process, so this process is studied as needed. In order to carry through this program it is necessary to study parabolic potential theory, that is, the potential theory of the heat equation, and the corresponding process of space time Brownian motion. No knowledge of potential theory is presupposed but it is assumed that the reader is familiar with basic probability concepts through conditional expectations. The necessary lattice theory, analytic set theory and capacity theory are covered in the Appendixes.
Thus this book on the one hand contains an introduction to classical and parabolic potential theory and on the other hand contains an introduc-
XXll Introduction
tion to martingale theory, including a smattering of the general theory of stochastic processes and of Markov process theory. There is cross referencing between the nonprobabilistic and probabilistic aspects of the work, and the linking of classical and parabolic potential theory with martingale theory, by Brownian motion and space time Brownian motion, is examined in depth.
One natural criticism of this project is that there is no reason to treat the very special potential theories of the Laplace and heat equations rather than general axiomatic potential theory. Another criticism is that there is no reason to treat potential theory other than as a special subhead of Markov process theory. In the author's opinion, however, classical potential theory is too important to serve merely as a source of illustrations of axiomatic potential theory, which theory in turn is too important in its own right to be left to the probabilists. To learn potential theory from probability is like learning algebraic geometry without the geometry.
It would be quite impossible to cover all those parts of modernized classical potential theory which are relevant to the purpose of this book. Thus there are striking gaps. For example the treatment of energy is skimpy, and Dirichlet spaces and the concept of bounded mean oscillation are not even mentioned in the text. The emphasis is on the Dirichlet problem and related topics; these are treated in considerable depth. The treatments of classical and parabolic potential theories are sometimes separated, sometimes together, but the notation is designed to exhibit the parallelism of the two theories: dots in the notation distinguish parabolic from classical concepts, thereby muddling eyes but saving brains. And the martingale theory notation is designed to point out to readers the corresponding potential theory notation.
Only the part of Markov process theory needed for the relevant discussion of Brownian motion and conditional Brownian motion is covered. In this book a stochastic process is a specified family of random variables, frequently coupled with a filtration to which the family is adapted, but the measure space of the process is left unspecified and there is no translation operator. Thus in a discussion of Brownian motion from a varying initial point the measure space on which the process is defined may vary with the initial point. This definition of a process may not be best for general Markov process theory but is convenient in the special context of this book; it implies for example that no matter how or on what measure space a process is defined, if it has the properties of a Brownian motion (continuous sample functions and the correct distributions of independent increments) then it is a Brownian motion. In a traditional song, a child finds an object which looks smells and tastes like a peanut so the child concludes that the object is a peanut. As stochastic processes are sometimes defined, with special properties demanded of the measure space on which the process random variables are defined, this simple logic is invalid. However the point of view of this book makes it essential in discussing Brownian motion to prove certain invariance properties, for example that two Brownian motion pro-
Introduction xxiii
cesses in N space, with a common initial point and variance parameter, have the same probability of hitting an analytic set. This fact is not trivial and such questions are treated.
There is nothing very novel in this book. Potential theorists may find the treatment of reductions on boundary sets of interest, as well as the use of iterated reductions to obtain limit theorems. Correspondingly, probabilists may find the new supermartingale crossing inequalities and the technique of iterated reductions of supermartingales of interest. A new domination principle for supermartingales illustrates the fact that classical potential theory still suggests interesting probability results.
The author thanks Bruce Hajek, Naresh Jain and John Taylor for helpful comments on various chapters and, finally, thanks his typist: usually faithful, sometimes accurate.
Notation and Conventions
[RN is N dimensional Euclidean space, [R = [Rl, and [R+ is the set [0, + oo[ of positive reals. lR is the set [ - 00, + 00] of extended reals and lR+ is the set [0, + 00] of positive extended reals.
7L is the set of integers, 7L+ is the set 0, 1, 2, ... , and 7L: is the set 0, ... , n. The boundary of an unbounded subset of [RN contains the adjoined point
00 of the one point compactification of [RN unless some other compactification has been specified. This boundary relative to the one point compactification of [RN will be called the Euclidean boundary of the set.
If ~ is a point of [RN and A is a subset of [RN the distance between ~ and A is written 1 ~ - A I.
B(~, <5) is the ball, in whatever metric space is specified, of center ~ and radius <5, specifically in [RN: B(~, <5) = {1]: 11] - ~I < <5}.
IN refers to N dimensional Lebesgue measure. If A and B are subsets of a space the set of points in A but not in B is
denoted by A-B. "Positive" means" 2 0" and monotone concepts are to be taken in the
wide sense, so that for example a constant function from [R into [R is both monotone increasing and monotone decreasing.
If D is an open subset of [RN the notation l[(k)(D) refers to the class of functions from D into [R which are continuous together with their derivatives of order :s::k.
Limit concepts for a function / at a point do not involve the value of/ at the point. Thus limq--+~/(1]) = rt. means that / is near rt. in small deleted neighborhoods of~.
The notation for a sequence frequently uses a dot for the index set; unless otherwise identified the index set is 7L + , so that A. = {Ao, AI, ... }.
The set on which a function/satisfies some set S of conditions is frequently denoted by {S}. Thus if/is a function from [R into [R the positivity set of/ is {f2 O}.
The book is divided into three Parts. Section 1.IIJ is Section 3 of Chapter II of Part 1 ; in any Part, Section II.3 is Section 3 of Chapter II of that part; in any Chapter, Section 3 is Section 3 of that Chapter, and so on.
