Classical Potential Theory and Its Probabilistic Counterpart: Advanced ProblemsPotential 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 withjaun diced 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 super martingales; 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. |
Contents
Chapter I | 3 |
Function Averages | 4 |
MaximumMinimum Theorem for Harmonic Functions | 5 |
The Fundamental Kernel for RN and Its Potentials | 6 |
Gauss Integral Theorem | 7 |
The Smoothness of Potentials The Poisson Equation | 8 |
Harmonic Measure and the Riesz Decomposition | 11 |
Chapter VII | 12 |
Optional Stopping | 442 |
Conditional Maximal Inequalities | 444 |
Crossings | 445 |
Forward Convergence in the L¹ Bounded Case | 450 |
Convergence of a Uniformly Integrable Martingale | 451 |
Forward Convergence of a Right Closable Supermartingale | 453 |
Backward Convergence of a Martingale | 454 |
Backward Convergence of a Supermartingale | 455 |
Chapter II | 14 |
Harnacks Inequality | 16 |
Convergence of Directed Sets of Harmonic Functions | 17 |
Harmonic Subharmonic and Superharmonic Functions | 18 |
Minimum Theorem for Superharmonic Functions | 20 |
Application of the Operation T | 21 |
Characterization of Superharmonic Functions in Terms of Harmonic Functions | 22 |
Differentiable Superharmonic Functions 9 Application of Jensens Inequality | 23 |
Superharmonic Functions on an Annulus | 24 |
Examples | 25 |
The Kelvin Transformation N | 26 |
Chapter IX | 27 |
The Fatou Boundary Limit Theorem | 31 |
Fundamental Convergence Theorem Preliminary Version | 37 |
The Natural Pointwise Order Decomposition for Positive Superharmonic | 43 |
Uniqueness of the Measure Determining a Potential | 50 |
Chapter V | 57 |
Greenian Sets in R2 as the Complements of Nonpolar Sets | 63 |
Green Functions | 66 |
Further Properties of | 90 |
Chapter VIII | 98 |
Examples | 104 |
Properties of PWB Solutions | 110 |
12 | 123 |
hHarmonic Measure u as a Function of D | 131 |
Interpretation of as a Green Function with Pole N | 140 |
Lattices and Related Classes of Functions | 141 |
The Class Dµ³½ | 142 |
The Class Lºµ½ p 1 | 144 |
The Lattices S and S | 145 |
The Vector Lattice S | 146 |
The Vector Lattice Sm | 148 |
The Vector Lattice Sp | 149 |
The Vector Lattice S qb | 150 |
The Vector Lattice S | 151 |
A Refinement of the Riesz Decomposition | 152 |
Chapter X | 155 |
Relation between Harmonic Measure and the Sweeping Kernel | 157 |
Sweeping Symmetry Theorem | 158 |
Swept Measures and Functions | 160 |
Some Properties of 84 | 161 |
Poles of a Positive Harmonic Function | 163 |
Relative Harmonic Measure on a Polar Set | 164 |
Chapter XI | 166 |
A Thinness Criterion | 168 |
Conditions That EA | 170 |
An Internal Limit Theorem | 171 |
Extension of the Fine Topology to RU 0 | 175 |
The Fine Topology Derived Set of a Subset of Rˇ | 177 |
Fine Topology Limits and Euclidean Topology Limits | 178 |
Fine Topology Limits and Euclidean Topology Limits Continued | 179 |
Identification of A¹ in Terms of a Special Function u | 180 |
Regularity in Terms of the Fine Topology | 181 |
The Euclidean Boundary Set of Thinness of a Greenian Set | 182 |
The Support of a Swept Measure | 183 |
A Special Reduction | 184 |
The Support of a Swept Measure Continuation of Section 14 | 185 |
Superharmonic Functions on FineOpen Sets | 187 |
Limits of Superharmonic Functions at Irregular Boundary Points of Their | 190 |
The Martin Functions | 196 |
Reductions on the Set of Minimal Martin Boundary Points | 203 |
The MinimalFine Topology | 210 |
MinimalFine Topology Limits and Martin Topology Limits at a Minimal | 216 |
Nontangential and MinimalFine Limits at a Halfspace Boundary | 222 |
Charges and Their Energies | 228 |
Alternative Proofs of Theorem 7b | 235 |
Sharpening of Lemma 4 | 237 |
Inner and Outer Capacities Notation of Section 10 | 240 |
Extremal Property Characterizations of Equilibrium Potentials Notation of Section 10 | 241 |
Expressions for CA | 243 |
The Gauss Minimum Problems and Their Relation to Reductions | 244 |
Dependence of C on D | 247 |
Energy Relative to R² | 248 |
The Wiener Thinness Criterion | 249 |
The Robin Constant and Equilibrium Measures Relative to R2 N 2 | 251 |
Chapter XIV | 256 |
Smoothness Properties of Superharmonic and Subharmonic Functions | 257 |
Green Functions | 258 |
Potentials of Measures | 259 |
Riesz Decomposition | 260 |
The Martin Boundary | 261 |
Chapter XV | 262 |
The Parabolic and Coparabolic Operators | 263 |
Coparabolic Polynomials | 264 |
The Parabolic Green Function of R¹ | 266 |
MaximumMinimum Parabolic Function Theorem | 267 |
Application of Greens Theorem | 269 |
The Parabolic Green Function of a Smooth Domain The Riesz Decom position and Parabolic Measure Formal Treatment | 270 |
The Green Function of an Interval | 272 |
Parabolic Measure for an Interval | 273 |
Parabolic Averages | 275 |
Harnacks Theorems in the Parabolic Context | 276 |
Superparabolic Functions | 277 |
Superparabolic Function Minimum Theorem | 279 |
The Operation tg and the Defining Average Properties of Superparabolic Functions | 280 |
Superparabolic and Parabolic Functions on a Cylinder | 281 |
The Appell Transformation | 282 |
Extensions of a Parabolic Function Defined on a Cylinder | 283 |
Contents xi | 285 |
A Generalized Superparabolic Function Inequality | 287 |
A Criterion of a Subparabolic Function Supremum | 288 |
A Condition that a Positive Parabolic Function Be Representable by a Poisson Integral | 290 |
The Parabolic Boundary Limit Theorem | 292 |
Minimal Parabolic Functions on a Slab | 293 |
Chapter XVII | 295 |
The Parabolic Context Reduction Operations | 296 |
The Parabolic Green Function | 298 |
Potentials | 300 |
The Smoothness of Potentials | 303 |
Riesz Decomposition Theorem | 305 |
The ParabolicFine Topology | 308 |
Semipolar Sets | 309 |
Preliminary List of Reduction Properties | 310 |
A Criterion of Parabolic Thinness | 313 |
The Parabolic Fundamental Convergence Theorem | 314 |
Applications of the Fundamental Convergence Theorem to Reductions and to Green Functions | 316 |
Applications of the Fundamental Convergence Theorem to the Parabolic Fine Topology | 317 |
Proofs of the Reduction Properties in Section 16 | 320 |
The Classical Context Green Function in Terms of the Parabolic Context Green Function N 1 | 326 |
The QuasiLindelöf Property | 328 |
Chapter XVIII | 329 |
hParabolic Measure | 332 |
Parabolic Barriers | 333 |
Relations between the Classical Dirichlet Problem and the Parabolic Context Dirichlet Problem | 334 |
Classical Reductions in the Parabolic Context | 335 |
Parabolic Regularity of Boundary Points | 337 |
Parabolic Regularity in Terms of the Fine Topology | 341 |
The Extension Ġ of Ġ and the Parabolic Average Ġ n when DCB | 343 |
Conditions that Є APS | 345 |
Parabolic and CoparabolicPolar Sets | 347 |
Parabolic and CoparabolicSemipolar Sets | 348 |
The Support of a Swept Measure | 350 |
An Internal Limit Theorem The CoparabolicFine Topology Smoothness of Superparabolic Functions | 351 |
Application to a Version of the Parabolic Context Fatou Boundary Limit Theorem on a Slab | 357 |
The Parabolic Context Domination Principle | 358 |
Martin Flat Point Set Pairs | 361 |
Chapter XIX | 363 |
The Martin Functions of Martin Point Set and Measure Set Pairs | 364 |
The Martin Space DM | 366 |
Preparatory Material for the Parabolic Context Martin Representation Theorem | 367 |
Minimal Parabolic Functions and Their Poles | 369 |
The Set of Nonminimal Martin Boundary Points | 370 |
The Martin Representation in the Parabolic Context | 371 |
Martin Boundaries for the Lower Halfspace of RN and for RN | 374 |
The Martin Boundary of D 0 + x 8 | 375 |
PWB Solutions on ĎM | 377 |
Boundary Counterpart of Theorem XVIII 14f | 379 |
The Vanishing of Potentials on MD | 381 |
Part 2 | 385 |
Chapter I | 387 |
Progressive Measurability | 388 |
Random Variables | 390 |
Contents xiii | 391 |
Conditional Expectation Continuity Theorem | 393 |
Fatous Lemma for Conditional Expectations | 396 |
Dominated Convergence Theorem for Conditional Expectations | 397 |
Stochastic Processes Evanescent Indistinguishable Standard Modi fication Nearly | 398 |
The Hitting of Sets and Progressive Measurability | 401 |
Canonical Processes and FiniteDimensional Distributions | 402 |
Choice of the Basic Probability Space | 404 |
The Hitting of Sets by a Right Continuous Process | 405 |
Measurability versus Progressive Measurability of Stochastic Processes | 409 |
Predictable Families of Functions | 410 |
Chapter II | 413 |
Optional Time Properties Continuous Parameter Context | 415 |
Process Functions at Optional Times | 417 |
Hitting and Entry Times | 419 |
Application to Continuity Properties of Sample Functions | 421 |
Continuation of Section 5 | 422 |
Predictable Optional Times | 423 |
Section Theorems | 425 |
The Graph of a Predictable Time and the Entry Time of a Predictable Set | 426 |
Semipolar Subsets of R+ Q | 427 |
The Classes D and LP of Stochastic Processes | 428 |
Decomposition of Optional Times Accessible and Totally Inaccessible Optional Times | 429 |
Chapter III | 432 |
Examples | 433 |
Elementary Properties Arbitrary Simply Ordered Parameter Set | 435 |
The Parameter Set in Martingale Theory | 437 |
Optional Sampling Theorem Bounded Optional Times | 438 |
Optional Sampling Theorem for Right Closed Processes | 440 |
The Natural Order Decomposition Theorem for Supermartingales | 457 |
The Operators LM and GM | 458 |
Supermartingale Potentials and the Riesz Decomposition | 459 |
Application to the Crossing Inequalities | 461 |
Chapter IV | 463 |
Optional Sampling of Uniformly Integrable Continuous Parameter Martingales | 468 |
Optional Sampling and Convergence of Continuous Parameter Supermartingales | 470 |
Increasing Sequences of Supermartingales | 473 |
Probability Version of the Fundamental Convergence Theorem of Potential Theory | 476 |
QuasiBounded Positive Supermartingales Generation of Supermartingale Potentials by Increasing Processes | 480 |
Natural versus Predictable Increasing Processes I Z or R | 483 |
Generation of Supermartingale Potentials by Increasing Processes in the Discrete Parameter Case | 488 |
An Inequality for Predictable Increasing Processes | 489 |
Generation of Supermartingale Potentials by Increasing Processes for Arbitrary Parameter Sets | 490 |
The Meyer Decomposition | 493 |
Meyer Decomposition of a Submartingale | 495 |
Role of the Measure Associated with a Supermartingale The Supermartingale Domination Principle | 496 |
The Operators τ LM and GM in the Continuous Parameter Context | 500 |
Potential Theory on R+ N | 501 |
The Fine Topology of R | 502 |
Potential Theory Reductions in a Continuous Parameter Probability Context | 504 |
Reduction Properties | 505 |
Proofs of the Reduction Properties in Section 18 | 509 |
Evaluation of Reductions | 513 |
The Energy of a Supermartingale Potential | 515 |
The Subtraction of a Supermartingale Discontinuity | 516 |
Supermartingale Decompositions and Discontinuities | 518 |
Chapter V | 520 |
x when x F Is a Submartingale | 521 |
Contents XV | 523 |
LP Bounded Stochastic Processes p 1 | 524 |
The Lattices S S S S | 525 |
The Vector Lattices S and S | 528 |
The Vector Lattices Sm and Sm | 529 |
The Vector Lattices S and Sp | 530 |
The Vector Lattices Sqb and Sqb | 531 |
The Vector Lattices S and S | 532 |
The Orthogonal Decompositions Sm Smqb + Sms and Sm Smgb + Sms | 533 |
Local Martingales and Singular Supermartingale Potentials in S | 534 |
Quasimartingales Continuous Parameter Context | 535 |
Chapter VI | 539 |
Choice of Filtration | 544 |
Integral Parameter Markov Processes with Stationary Transition Proba bilities | 545 |
Application of Martingale Theory to Discrete Parameter Markov Processes | 547 |
Continuous Parameter Markov Processes with Stationary Transition Probabilities | 550 |
Specialization to Right Continuous Processes | 552 |
Lifetimes and Trap Points | 554 |
Right Continuity of Markov Process Filtrations A ZeroOne 01 Law | 556 |
Strong Markov Property | 557 |
Probabilistic Potential Theory Excessive Functions | 560 |
Excessive Functions and Supermartingales | 564 |
Excessive Functions and the Hitting Times of Analytic Sets Notation and Hypotheses of Section 11 | 565 |
Conditioned Markov Processes | 566 |
Tied Down Markov Processes | 567 |
Killed Markov Processes | 568 |
Chapter VII | 570 |
Brownian Motion | 572 |
Continuity of Brownian Paths | 576 |
Brownian Motion Filtrations | 578 |
Elementary Properties of the Brownian Transition Density and Brownian Motion | 581 |
The ZeroOne Law for Brownian Motion | 583 |
Tied Down Brownian Motion | 586 |
André Reflection Principle | 587 |
Brownian Motion in an Open Set N 1 | 589 |
SpaceTime Brownian Motion in an Open Set | 592 |
Brownian Motion in an Interval | 594 |
Probabilistic Evaluation of Parabolic Measure for an Interval | 595 |
Probabilistic Significance of the Heat Equation and Its Dual | 596 |
Chapter VIII | 599 |
The Size of To | 601 |
Properties of the Itô Integral | 602 |
The Stochastic Integral for an Integrand Process in To | 605 |
Proofs of the Properties in Section 3 | 607 |
Extension to VectorValued and ComplexValued Integrands | 611 |
Martingales Relative to Brownian Motion Filtrations | 612 |
A Change of Variables | 615 |
The Role of Brownian Motion Increments | 618 |
N1 Computation of the Itô Integral by RiemannStieltjes Sums | 620 |
Itôs Lemma | 621 |
The Composition of the Basic Functions of Potential Theory with Brownian Motion | 625 |
The Composition of an Analytic Function with Brownian Motion | 626 |
Brownian Motion and Martingale Theory | 627 |
Coparabolic Polynomials and Martingale Theory | 630 |
Superharmonic and Harmonic Functions on RN and Supermartingales and Martingales | 632 |
Hitting of an F Set | 635 |
The Hitting of a Set by Brownian Motion | 636 |
Superharmonic Functions Excessive for Brownian Motion | 637 |
Preliminary Treatment of the Composition of a Superharmonic Function with Brownian Motion A Probabilistic Fatou Boundary Limit Theorem | 641 |
Excessive and Invariant Functions for Brownian Motion | 645 |
Application to Hitting Probabilities and to Parabolicity of Transition Densities | 647 |
N 2 The Hitting of Nonpolar Sets by Brownian Motion | 648 |
Continuity of the Composition of a Function with Brownian Motion | 649 |
Continuity of Superharmonic Functions on Brownian Motion | 650 |
Preliminary Probabilistic Solution of the Classical Dirichlet Problem | 651 |
Probabilistic Evaluation of Reductions | 653 |
Probabilistic Description of the Fine Topology | 656 |
aExcessive Functions for Brownian Motion and Their Composition with Brownian Motions | 659 |
Brownian Motion Transition Functions as Green Functions The Corre sponding Backward and Forward Parabolic Equations | 661 |
Excessive Measures for Brownian Motion | 663 |
Nearly Borel Sets for Brownian Motion | 666 |
Conditional Brownian Motion | 668 |
hBrownian Motion in Terms of Brownian Motion | 671 |
Contexts for 2 1 | 676 |
Asymptotic Character of hBrownian Paths at Their Lifetimes | 677 |
hBrownian Motion from an Infinity of h | 680 |
Brownian Motion under Time Reversal | 682 |
Preliminary Probabilistic Solution of the Dirichlet Problem for hHarmonic Functions hBrownian Motion Hitting Probabilities and the Corresponding ... | 684 |
Probabilistic Boundary Limit and Internal Limit Theorems for Ratios of Strictly Positive Superharmonic Functions | 688 |
Conditional Brownian Motion in a Ball | 691 |
Conditional Brownian Motion Last Hitting Distributions The Capacitary Distribution of a Set in Terms of a Last Hitting Distribution | 693 |
The Tail σ Algebra of a Conditional Brownian Motion | 694 |
Conditional SpaceTime Brownian Motion | 699 |
SpaceTime Brownian Motion in R RN with Parameter Set R | 700 |
Part 3 | 703 |
Chapter I | 705 |
Relations between Decomposition Components of S in Potential Theory and Martingale Theory | 706 |
PWBRelated Conditions on hHarmonic Functions and on Martingales | 707 |
Class D Property versus QuasiBoundedness | 708 |
A Condition for QuasiBoundedness | 709 |
Singularity of an Element of S | 710 |
The Singular Component of an Element of S | 711 |
The Class Spqb | 712 |
The Class Sps | 714 |
Lattice Theoretic Analysis of the Composition of an hSuperharmonic Function with an hBrownian Motion | 715 |
A Decomposition of S Potential Theory Context | 716 |
Continuation of Section 11 | 717 |
Chapter II | 719 |
Probabilistic Analysis of the PWB Method | 720 |
PWB Examples | 723 |
Tail o Algebras in the PWB Context | 725 |
Chapter III | 727 |
Brownian Motions from Martin Boundary Points Notation of Section 1 | 728 |
The ZeroOne Law at a Minimal Martin Boundary Point and the Probabilistic Formulation of the MinimalFine Topology Notation of Section 1 | 730 |
The Probabilistic Fatou Theorem on the Martin Space | 732 |
Probabilistic Approach to Theorem 1 XI 4c and Its Boundary Counterparts | 733 |
Martin Representation of Harmonic Functions in the Parabolic Context | 735 |
Appendixes | 739 |
Appendix I | 741 |
Sets Analytic over a Product Paving | 742 |
Analytic Extensions versus σ Algebra Extensions of Pavings | 743 |
The Operation A | 744 |
Extension of a Measurability Concept to the Analytic Operation Context | 745 |
Polish Spaces | 746 |
Analytic Sets | 747 |
Analytic Subsets of Polish Spaces | 748 |
Appendix II | 750 |
Choquet Capacity Theorem | 751 |
A Fundamental Example of a Choquet Capacity | 752 |
Generation of a Choquet Capacity by a Positive Strongly Subadditive Set Function | 753 |
Topological Precapacities | 755 |
Universally Measurable Sets | 756 |
Lattice Theory | 758 |
The Specific Order Generated by a Cone | 759 |
Vector Lattices | 760 |
Decomposition Property of a Vector Lattice | 762 |
Projections on Bands | 763 |
The Orthogonal Complement of a Set | 764 |
Order Convergence | 765 |
Order Convergence on a Linearly Ordered Set | 766 |
Appendix IV | 767 |
Composition of Functions | 768 |
The Measure Lattice of a Measurable Space | 769 |
The Finite Measure Lattice of a Measurable Space Notation of Section 4 | 771 |
The Hahn and Jordan Decompositions | 772 |
Absolute Continuity and Singularity | 773 |
Lattices of Measurable Functions on a Measure Space | 774 |
Order Convergence of Families of Measurable Functions | 775 |
Measures on Polish Spaces | 777 |
Derivates of Measures | 778 |
Appendix V | 779 |
Appendix VI | 781 |
Universally Measurable Extension of a Kernel | 782 |
Appendix VII | 785 |
Ratio Integral Limit Theorems | 786 |
A Ratio Integral Limit Theorem Involving Convex Variational Derivates | 788 |
Lower Semicontinuous Functions | 791 |
Part 2 | 806 |
Bibliography | 819 |
| 823 | |
Other editions - View all
Common terms and phrases
According to Section According to Theorem algebra Application arbitrary ball Borel measurable boundary function boundary limit boundary subsets bounded Brownian motion classical context condition coparabolic countable D₁ defined equality equation Euclidean boundary everywhere example fact finite valued follows Green function Greenian set Greenian subset h-Brownian h-harmonic h-resolutive h₁ harmonic function harmonic measure Hence identically implies increasing sequence inequality infimum Lemma lower semicontinuous Markov process Martin boundary Martin boundary point measure null metric minimal minimal-fine neighborhood notation null sets Observe open subset optional parabolic context parabolic function parabolic-fine parameter set polar set positive harmonic function positive superharmonic function potential theory proof prove random variable restriction sample function satisfied smoothed reduction subharmonic subset of RN supermartingale superparabolic function suppose surely right continuous topology transition function trivial true u₁ u₂ uniformly integrable upper PWB vanishes