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140122  eng 
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a 9783642565731

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1 

a Doob, Joseph L.

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a Classical Potential Theory and Its Probabilistic Counterpart
h Elektronische Ressource
c by Joseph L. Doob

250 


a 1st ed. 2001

260 


a Berlin, Heidelberg
b Springer Berlin Heidelberg
c 2001, 2001

300 


a L, 1551 p
b online resource

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a I Introduction to the Mathematical Background of Classical Potential Theory  II Basic Properties of Harmonic, Subharmonic, and Superharmonic Functions  III Infima of Families of Superharmonic Functions  IV Potentials on Special Open Sets  V Polar Sets and Their Applications  VI The Fundamental Convergence Theorem and the Reduction Operation  VII Green Functions  VIII The Dirichlet Problem for Relative Harmonic Functions  IX Lattices and Related Classes of Functions  X The Sweeping Operation  XI The Fine Topology  XII The Martin Boundary  XIII Classical Energy and Capacity  XIV OneDimensional Potential Theory  XV Parabolic Potential Theory: Basic Facts  XVI Subparabolic, Superparabolic, and Parabolic Functions on a Slab  XVII Parabolic Potential Theory (Continued)  XVIII The Parabolic Dirichlet Problem, Sweeping, and Exceptional Sets  XIX The Martin Boundary in the Parabolic Context  I Fundamental Concepts of Probability 

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a II Optional Times and Associated Concepts  III Elements of Martingale Theory  IV Basic Properties of Continuous Parameter Supermartingales  V Lattices and Related Classes of Stochastic Processes  VI Markov Processes  VII Brownian Motion  VIII The Itô Integral  IX Brownian Motion and Martingale Theory  X Conditional Brownian Motion  I Lattices in Classical Potential Theory and Martingale Theory  II Brownian Motion and the PWB Method  III Brownian Motion on the Martin Space  Appendixes  Appendix I  Analytic Sets  1. Pavings and Algebras of Sets  2. Suslin Schemes  3. Sets Analytic over a Product Paving  4. Analytic Extensions versus ? Algebra Extensions of Pavings  7. Projections of Sets in Product Pavings  8. Extension of a Measurability Concept to the Analytic Operation Context  10. Polish Spaces  11. The Baire Null Space  12. Analytic Sets  13. Analytic Subsets of Polish Spaces  Appendix II  Capacity Theory 

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a 1. Choquet Capacities  2. Sierpinski Lemma  3. Choquet Capacity Theorem  4. Lusin’s Theorem  5. A Fundamental Example of a Choquet Capacity  6. Strongly Subadditive Set Functions  7. Generation of a Choquet Capacity by a Positive Strongly Subadditive Set Function  8. Topological Precapacities  9. Universally Measurable Sets  Appendix III  Lattice Theory  1. Introduction  2. Lattice Definitions  3. Cones  4. The Specific Order Generated by a Cone  5. Vector Lattices  6. Decomposition Property of a Vector Lattice  7. Orthogonality in a Vector Lattice  8. Bands in a Vector Lattice  9. Projections on Bands  10. The Orthogonal Complement of a Set  11. The Band Generated by a Single Element  12. Order Convergence  13. Order Convergence on a Linearly Ordered Set  Appendix IV  Lattice Theoretic Concepts in Measure Theory  1. Lattices of Set Algebras  2. Measurable Spaces and Measurable Functions  3. Composition of Functions 

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a 3. Choquet Topological Lemma  Historical Notes  1  2  3  Appendixes  Notation Index

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a 4. The Measure Lattice of a Measurable Space  5. The ? Finite Measure Lattice of a Measurable Space (Notation of Section 4)  6. The Hahn and Jordan Decompositions  8. Absolute Continuity and Singularity  9. Lattices of Measurable Functions on a Measure Space  10.Order Convergence of Families of Measurable Functions  11. Measures on Polish Spaces  12. Derivates of Measures  Appendix V  Uniform Integrability  Appendix VI  Kernels and Transition Functions  1. Kernels  2. Universally Measurable Extension of a Kernel  3. Transition Functions  Appendix VII  Integral Limit Theorems  1. An Elementary Limit Theorem  2. Ratio Integral Limit Theorems  3. A OneDimensional Ratio Integral Limit Theorem  4. A Ratio Integral Limit Theorem Involving Convex Variational Derivates  Appendix VIII  Lower Semicontinuous Functions  1. The Lower Semicontinuous Smoothing of a Function  2. Suprema of Families of Lower Semicontinuous Functions 

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a Potential theory (Mathematics)

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a Potential Theory

653 


a Probability Theory and Stochastic Processes

653 


a Probabilities

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2 

a SpringerLink (Online service)

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0 
7 
a eng
2 ISO 6392

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b SBA
a Springer Book Archives 2004

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a Classics in Mathematics

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u https://doi.org/10.1007/9783642565731?nosfx=y
x Verlag
3 Volltext

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a 515.96

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a From the reviews: "This huge book written in several years by one of the few mathematicians able to do it, appears as a precise and impressive study (not very easy to read) of this bothsided question that replaces, in a coherent way, without being encyclopaedic, a large library of books and papers scattered without a uniform language. Instead of summarizing the author gives his own way of exposition with original complements. This requires no preliminary knowledge. ...The purpose which the author explains in his introduction, i.e. a deep probabilistic interpretation of potential theory and a link between two great theories, appears fulfilled in a masterly manner". M. Brelot in Metrika (1986)
