Classical potential theory and its probabilistic counterpart:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Berlin [u.a.]
Springer
2001
|
| Ausgabe: | Reprint of the 1984 ed. |
| Schriftenreihe: | Classics in mathematics
|
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | Ursprüngl. als: Grundlehren der mathematischen Wissenschaften; 262. |
| Beschreibung: | XXIII, 846 S. |
| ISBN: | 3540412069 9783540412069 |
Internformat
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| 100 | 1 | |a Doob, Joseph L. |d 1910-2004 |e Verfasser |0 (DE-588)12778022X |4 aut | |
| 245 | 1 | 0 | |a Classical potential theory and its probabilistic counterpart |c J. L. Doob |
| 250 | |a Reprint of the 1984 ed. | ||
| 264 | 1 | |a Berlin [u.a.] |b Springer |c 2001 | |
| 300 | |a XXIII, 846 S. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 0 | |a Classics in mathematics | |
| 500 | |a Ursprüngl. als: Grundlehren der mathematischen Wissenschaften; 262. | ||
| 650 | 4 | |a Potential theory (Mathematics) | |
| 650 | 4 | |a Harmonic functions | |
| 650 | 4 | |a Martingales (Mathematics) | |
| 650 | 0 | 7 | |a Wahrscheinlichkeitsrechnung |0 (DE-588)4064324-4 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Martingaltheorie |0 (DE-588)4168982-3 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Martingal |0 (DE-588)4126466-6 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Stochastischer Prozess |0 (DE-588)4057630-9 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Potenzialtheorie |0 (DE-588)4046939-6 |2 gnd |9 rswk-swf |
| 689 | 0 | 0 | |a Potenzialtheorie |0 (DE-588)4046939-6 |D s |
| 689 | 0 | 1 | |a Martingaltheorie |0 (DE-588)4168982-3 |D s |
| 689 | 0 | |5 DE-604 | |
| 689 | 1 | 0 | |a Potenzialtheorie |0 (DE-588)4046939-6 |D s |
| 689 | 1 | 1 | |a Martingal |0 (DE-588)4126466-6 |D s |
| 689 | 1 | 2 | |a Wahrscheinlichkeitsrechnung |0 (DE-588)4064324-4 |D s |
| 689 | 1 | 3 | |a Stochastischer Prozess |0 (DE-588)4057630-9 |D s |
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Datensatz im Suchindex
| _version_ | 1804128262198132736 |
|---|---|
| adam_text | Contents
Introduction
xxi
Notation and Conventions xxv
Parti
Classical and Parabolic Potential Theory
Chapter I
Introduction to the Mathematical Background of Classical Potential
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 UN 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
.....................
Π
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
τΒ
.................................... 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
(μΒ_)
and D(/íb_)
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
ILS)
.................................................... 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
Vil
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
ł
.
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
ΰομ
of a Measure
μ
.................................. 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-ki 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 =s
1)___ 106
5.
A-Harmonic Measure Null Sets
..................................... 108
6.
Properties of PWB* Solutions
...................................... 110
7.
Proofs for Section
6..............................................
lil
8.
A-Harmonic Measure
............................................. 114
9.
A-Resolutive Boundaries
........................................... 118
10.
Relations between Reductions and Dirichlet Solutions
................. 122
11.
Generalization of the Operator
τ£
and Application to GM*
............. 123
12.
Barriers
......................................................... 124
13.
А
-Barriers and Boundary Point A-Regularity
.......................... 126
14.
Barriers and Euclidean Boundary Point Regularity
.................... 127
15.
The Geometrical Significance of Regularity {Euclidean Boundary, A
= 1). 128
16.
Continuation of Section
13........................................ 130
17.
A-Harmonic Measure
μ%
as a Function of
D
.......................... 131
18.
The Extension
G¡
of GD and the Harmonic Average
μο(ζ^ΐ(η,·))
When
D
cz
В
.......................................................... 132
19.
Modification of Section
18
for
D = R2 ..............................
136
20.
Interpretation of
фо
as a Green Function with Pole oo (N
= 2).......... 139
21.
Variant of the Operator
τΒ
......................................... 140
viii Contents
Chapter IX
Lattices and Related Classes of Functions
....................... 141
1.
Introduction
..................................................... 141
2.
LMţ,
и
for an A-Subharmonic Function
и
............................ 141
3.
The Class D(^-)
................................................
142
4.
The Class I/OiJ,.) (p
> 1)......................................... 144
5.
The Lattices
(Ѕ±,
<)
and
(S+;í<)
................................... 145
6.
The Vector Lattice (S,
<) ......................................... 146
7.
The Vector Lattice Sm
............................................. 148
8.
The Vector Lattice Sp
............................................. 149
9.
The Vector Lattice S,b
............................................ 150
10.
The Vector Lattice S5
............................................. 151
11.
A Refinement of the Riesz Decomposition
........................... 152
12.
Lattices of
й-Нагтопіс
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
<5¿
............................................. 158
5.
Swept Measures and Functions
..................................... 160
6.
Some Properties of
<5¿
............................................. 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
ţeAf
............................................ 169
4.
An Internal Limit Theorem
........................................ 171
5.
Extension of the Fine Topology to
R u
{oo}
......................... 175
6.
The Fine Topology Derived Set of a Subset of R
..................... 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 Af in Terms of a Special Function u*
................. 180
11. Quasi-Lindelöf
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
μ Λ
.......................................... 183
16.
A Special Reduction
.............................................. 184
17.
The Fine Interior of a Set of Constancy of
a Superharmonie
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
!μΏ
.................................. 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
R
.................... 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
Dt-tGDß............................................ 230
6.
Classical Evaluation of Energy; Hubert Space Methods
................ 231
7.
The Energy Functional (Relative to an Arbitrary Greenian Subset
D
of
W)
............................................................. 233
8.
Alternative Proofs of Theorem
7(è+)
................................ 235
X
Contents
9.
Sharpening of Lemma
4........................................... 237
10.
The Classical Capacity Function
................................... 237
11.
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{Ä)............................................. 243
14.
The Gauss Minimum Problems and Their Relation to Reductions
....... 244
15.
Dependence of C* on
D
........................................... 247
16.
Energy Relative to R2
............................................. 248
17.
The Wiener Thinness Criterion
..................................... 249
18.
The Robin Constant and Equilibrium Measures Relative to
IR2
(Ν =
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
Ř
................................ 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
i¿
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
Contents Xl
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
V-ifté-)
and Oifig-) 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
>
I)
........................................... 326
19.
The
Quasi-Lindelöf
Property
....................................... 328
Chapter
XVIII
The Parabolic Dirichlet Problem, Sweeping, and Exceptional Sets
... 329
ł
.
Relativizat
ion of the Parabolic Context
;
The PWB Method in this
Context
......................................................... 329
2.
A-Parabolie 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
Xli Contents
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
G¿
of
G¿
and the Parabolic Average
/¿¿(¿Łój
(·,»)))
when
Da
В
.......................................................... 343
10.
Conditions that
ξεΑρ/
............................................ 345
11.
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
ĎM
............................................ 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
Ď
=
W
x
]0, <5[ with
0 <
б
<, +
oo
......... 371
9.
Martin Boundaries for the Lower Half-space of
ŔN
and for
Ř
.......... 374
10.
The Martin Boundary of
Ď
=
]0, +oo[
x ]-ao,<5[...................
375
VI.
PWB* Solutions on ¿M
........................................... 377
12.
The Minimal-Fine Topology in the Parabolic Context
................. 377
13.
Boundary Counterpart of Theorem
XVIII.
14(f
)...................... 379
14.
The Vanishing of Potentials on
0MĎ
................................ 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
1.
Adapted Families of Functions on Measurable Spaces
................. 387
2.
Progressive Measurability
......................................... 388
3.
Random Variables
............................................... 390
Contents
ХШ
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
R+
χ Ω......................................
427
11.
The Classes
D
and
L
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
Ľ
Inequality for
Submartingale Suprema.........................
444
12.
Crossings
....................................................... 445
13.
Forward Convergence in the L1 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
xiv
Contents
17.
Backward Convergence of
a 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 of Supermartingale
Potentials by Increasing Processes
.................................. 480
7.
Natural versus Predictable Increasing Processes (I
—
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
τ,
LM, and GM in the Continuous Parameter Context.
... 500
15.
Potential Theory on R+
χ Ω.......................................
501
16.
The Fine Topology of R+
χ Ω.....................................
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.
LMjcí·)
when
{*(·),
^(·)}
Is
a Submartingale
........................ 521
Contents
XV
3.
Uniformly
Integrable
Positive
Submartingales
........................ 523
4.
U
Bounded Stochastic Processes (p
> 1)............................ 524
5.
The Lattices ( S*,
«£),
( S+, <0,
(S*,
£),
(S+,
S).....................
525
6.
The Vector Lattices ( S,
^)
and (S,
^)
.............................. 528
7.
The Vector Lattices ( S^
<)
and
(S„,
<)............................ 529
8.
The Vector Lattices ( Sp,
:<)
and (Sp,
<)............................. 530
9.
The Vector Lattices ( S,»,
<)
and (S,b,
<)........................... 531
10.
The Vector Lattices ( Ss,
<)
and (Ss,
<,)............................. 532
11.
The Orthogonal Decompositions Sm
=
Smqb
+
Ѕиѕ
and
Ѕш =
SL^
+
Sms
. 533
12.
Local Martingales and Singular
Supermartingale
Potentials in (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
11.
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 UN
............ 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.
André
Reflection Principle
......................................... 587
9.
Brownian Motion in an Open Set
(N ž
ł)............................
589
10.
Space-Time Brownian Motion in an Open Set
........................ 592
11.
Brownian Motion in an Interval
.................................... 594
xvi
Contents
12.
Probabilistic Evaluation of Parabolic Measure foran Interval
.......... 595
13.
Probabilistic Significance of the Heat Equation and Its Dual
......■..... 596
Chapter
VIII
The
Ito
Integral
.............................................. 599
1.
Notation
........................................................ 599
2.
The Size of
Го
................................................... 601
3.
Properties of the
Ito
Integral
....................................... 602
4.
The Stochastic Integral for an Integrand Process in
Го
................. 605
5.
The Stochastic Integral for an Integrand Process in
Γ
.................. 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.
Itô 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 R 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.
a-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
Borei
Sets for Brownian Motion
............................. 666
20.
Brownian Motion into a Set from an Irregular Boundary Point
......... 666
Contents XVII
Chapter
Χ
Conditional Brownian Motion
................................. 668
1.
Definition
....................................................... 668
2.
A-Brownian Motion in Terms of Brownian Motion
.................... 671
3.
Contexts for
(2.1)................................................ 676
4.
Asymptotic Character of A-Brownian Paths at Their Lifetimes
.......... 677
5.
A-Brownian Motion from an Infinity of A
............................ 680
6.
Brownian Motion under Time Reversal
............................. 682
7.
Preliminary Probabilistic Solution of the Dirichlet Problem for A-Harmonic
Functions; A-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
11.
The Tail
σ
Algebra of a Conditional Brownian Motion
................ 694
12.
Conditional Space-Time Brownian Motion
.......................... 699
13.
[Space-Time] Brownian Motion in
[Ŕ ]
UN with Parameter Set
IR
....... 700
Part3
Chapter I
Lattices in Classical Potential Theory and Martingale Theory
....... 705
1.
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 L and
D
............................................. 706
4.
PWB-Related Conditions on A-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 of S*
.................................... 710
8.
The Singular Component of an Element of S+
........................ 711
9.
The Class
Ѕт„
................................................... 712
10.
The Class Sps
.................................................... 714
11.
Lattice Theoretic Analysis of the Composition of an A-Superharmonic
Function with an A-Brownian Motion
............................... 715
12.
A Decomposition of S*s (Potential Theory Context)
................... 716
13.
Continuation of Section
11 ........................................ 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.
PWB* Examples
................................................. 723
4.
Tail
σ
Algebras in the PWB* Context
................................ 725
Chapter III
Brownian Motion on the Martin Space
.......................... 727
1.
The Structure of Brownian Motion on the Martin Space
............... 727
2.
Brownian Motions from Martin Boundary Points (Notation of Section
1) 728
3.
The Zero-One Law at a Minimal Martin Boundary Point and the
Probabilistic Formulation of the Minimal-Fine Topology (Notation of
Section
1)....................................................... 730
4.
The Probabilistic Fatou Theorem on the Martin Space
................. 732
5.
Probabilistic Approach to Theorem l.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
σ
Algebra Extensions of Pavings
............ 743
5.
Projection Characterization
ѕ/ЦШ)
.................................. 743
6.
The Operation
sł(sd}
............................................. 744
7.
Projections of Sets in Product Pavings
............................... 744
8.
Extension of a Measurability Concept to the Analytic Operation Context
. 745
9.
The Gd 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
11.
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
1.
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
σ
Finite Measure Lattice of a Measurable Space (Notation of Section
4) 771
6.
The
Hahn
and Jordan Decompositions
.............................. 772
7.
The Vector Lattice Jia
............................................ 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
11.
Measures on Polish Spaces
........................................ 777
12. Derivates
of Measures
............................................ 778
Appendix V
Uniform Integrability
........................................ 779
Appendix VI
Kernels and Transition Functions
.............................. 781
1.
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
VIH
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
|
| any_adam_object | 1 |
| author | Doob, Joseph L. 1910-2004 |
| author_GND | (DE-588)12778022X |
| author_facet | Doob, Joseph L. 1910-2004 |
| author_role | aut |
| author_sort | Doob, Joseph L. 1910-2004 |
| author_variant | j l d jl jld |
| building | Verbundindex |
| bvnumber | BV013469679 |
| callnumber-first | Q - Science |
| callnumber-label | QA404 |
| callnumber-raw | QA404.7.D66 2001 |
| callnumber-search | QA404.7.D66 2001 |
| callnumber-sort | QA 3404.7 D66 42001 |
| callnumber-subject | QA - Mathematics |
| classification_rvk | SK 540 SK 560 SK 820 |
| classification_tum | MAT 607f MAT 310f |
| ctrlnum | (OCoLC)76199780 (DE-599)BVBBV013469679 |
| dewey-full | 515/.921 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 515 - Analysis |
| dewey-raw | 515/.9 21 |
| dewey-search | 515/.9 21 |
| dewey-sort | 3515 19 221 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| edition | Reprint of the 1984 ed. |
| format | Book |
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| id | DE-604.BV013469679 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-09T18:46:27Z |
| institution | BVB |
| isbn | 3540412069 9783540412069 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-009194328 |
| oclc_num | 76199780 |
| open_access_boolean | |
| owner | DE-29T DE-83 DE-11 DE-355 DE-BY-UBR DE-91G DE-BY-TUM |
| owner_facet | DE-29T DE-83 DE-11 DE-355 DE-BY-UBR DE-91G DE-BY-TUM |
| physical | XXIII, 846 S. |
| publishDate | 2001 |
| publishDateSearch | 2001 |
| publishDateSort | 2001 |
| publisher | Springer |
| record_format | marc |
| series2 | Classics in mathematics |
| spelling | Doob, Joseph L. 1910-2004 Verfasser (DE-588)12778022X aut Classical potential theory and its probabilistic counterpart J. L. Doob Reprint of the 1984 ed. Berlin [u.a.] Springer 2001 XXIII, 846 S. txt rdacontent n rdamedia nc rdacarrier Classics in mathematics Ursprüngl. als: Grundlehren der mathematischen Wissenschaften; 262. Potential theory (Mathematics) Harmonic functions Martingales (Mathematics) Wahrscheinlichkeitsrechnung (DE-588)4064324-4 gnd rswk-swf Martingaltheorie (DE-588)4168982-3 gnd rswk-swf Martingal (DE-588)4126466-6 gnd rswk-swf Stochastischer Prozess (DE-588)4057630-9 gnd rswk-swf Potenzialtheorie (DE-588)4046939-6 gnd rswk-swf Potenzialtheorie (DE-588)4046939-6 s Martingaltheorie (DE-588)4168982-3 s DE-604 Martingal (DE-588)4126466-6 s Wahrscheinlichkeitsrechnung (DE-588)4064324-4 s Stochastischer Prozess (DE-588)4057630-9 s 1\p DE-604 Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009194328&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Doob, Joseph L. 1910-2004 Classical potential theory and its probabilistic counterpart Potential theory (Mathematics) Harmonic functions Martingales (Mathematics) Wahrscheinlichkeitsrechnung (DE-588)4064324-4 gnd Martingaltheorie (DE-588)4168982-3 gnd Martingal (DE-588)4126466-6 gnd Stochastischer Prozess (DE-588)4057630-9 gnd Potenzialtheorie (DE-588)4046939-6 gnd |
| subject_GND | (DE-588)4064324-4 (DE-588)4168982-3 (DE-588)4126466-6 (DE-588)4057630-9 (DE-588)4046939-6 |
| title | Classical potential theory and its probabilistic counterpart |
| title_auth | Classical potential theory and its probabilistic counterpart |
| title_exact_search | Classical potential theory and its probabilistic counterpart |
| title_full | Classical potential theory and its probabilistic counterpart J. L. Doob |
| title_fullStr | Classical potential theory and its probabilistic counterpart J. L. Doob |
| title_full_unstemmed | Classical potential theory and its probabilistic counterpart J. L. Doob |
| title_short | Classical potential theory and its probabilistic counterpart |
| title_sort | classical potential theory and its probabilistic counterpart |
| topic | Potential theory (Mathematics) Harmonic functions Martingales (Mathematics) Wahrscheinlichkeitsrechnung (DE-588)4064324-4 gnd Martingaltheorie (DE-588)4168982-3 gnd Martingal (DE-588)4126466-6 gnd Stochastischer Prozess (DE-588)4057630-9 gnd Potenzialtheorie (DE-588)4046939-6 gnd |
| topic_facet | Potential theory (Mathematics) Harmonic functions Martingales (Mathematics) Wahrscheinlichkeitsrechnung Martingaltheorie Martingal Stochastischer Prozess Potenzialtheorie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009194328&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT doobjosephl classicalpotentialtheoryanditsprobabilisticcounterpart |