Classical Potential Theory and Its Probabilistic Counterpart:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Elektronisch E-Book |
| Sprache: | English |
| Veröffentlicht: |
New York, NY
Springer New York
1984
|
| Schriftenreihe: | Grundlehren der mathematischen Wissenschaften, A Series of Comprehensive Studies in Mathematics
262 |
| Schlagworte: | |
| Online-Zugang: | Volltext |
| Beschreibung: | 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 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 |
| Beschreibung: | 1 Online-Ressource (XXVI, 847 p) |
| ISBN: | 9781461252085 9781461297383 |
| ISSN: | 0072-7830 |
| DOI: | 10.1007/978-1-4612-5208-5 |
Internformat
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Datensatz im Suchindex
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| any_adam_object | |
| author | Doob, J. L. |
| author_facet | Doob, J. L. |
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| author_sort | Doob, J. L. |
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| building | Verbundindex |
| bvnumber | BV042420372 |
| classification_tum | MAT 000 |
| collection | ZDB-2-SMA ZDB-2-BAE |
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| dewey-ones | 515 - Analysis |
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| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| doi_str_mv | 10.1007/978-1-4612-5208-5 |
| format | Electronic eBook |
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| id | DE-604.BV042420372 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-10T01:21:06Z |
| institution | BVB |
| isbn | 9781461252085 9781461297383 |
| issn | 0072-7830 |
| language | English |
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| publishDate | 1984 |
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| series2 | Grundlehren der mathematischen Wissenschaften, A Series of Comprehensive Studies in Mathematics |
| spelling | Doob, J. L. Verfasser aut Classical Potential Theory and Its Probabilistic Counterpart by J. L. Doob New York, NY Springer New York 1984 1 Online-Ressource (XXVI, 847 p) txt rdacontent c rdamedia cr rdacarrier Grundlehren der mathematischen Wissenschaften, A Series of Comprehensive Studies in Mathematics 262 0072-7830 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 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 Mathematics Potential theory (Mathematics) Distribution (Probability theory) Potential Theory Probability Theory and Stochastic Processes Mathematik Potenzialtheorie (DE-588)4046939-6 gnd rswk-swf Stochastischer Prozess (DE-588)4057630-9 gnd rswk-swf Martingal (DE-588)4126466-6 gnd rswk-swf Martingaltheorie (DE-588)4168982-3 gnd rswk-swf Wahrscheinlichkeitsrechnung (DE-588)4064324-4 gnd rswk-swf Potenzialtheorie (DE-588)4046939-6 s Martingal (DE-588)4126466-6 s Wahrscheinlichkeitsrechnung (DE-588)4064324-4 s Stochastischer Prozess (DE-588)4057630-9 s 1\p DE-604 Martingaltheorie (DE-588)4168982-3 s 2\p DE-604 https://doi.org/10.1007/978-1-4612-5208-5 Verlag Volltext 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 2\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Doob, J. L. Classical Potential Theory and Its Probabilistic Counterpart Mathematics Potential theory (Mathematics) Distribution (Probability theory) Potential Theory Probability Theory and Stochastic Processes Mathematik Potenzialtheorie (DE-588)4046939-6 gnd Stochastischer Prozess (DE-588)4057630-9 gnd Martingal (DE-588)4126466-6 gnd Martingaltheorie (DE-588)4168982-3 gnd Wahrscheinlichkeitsrechnung (DE-588)4064324-4 gnd |
| subject_GND | (DE-588)4046939-6 (DE-588)4057630-9 (DE-588)4126466-6 (DE-588)4168982-3 (DE-588)4064324-4 |
| 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 by J. L. Doob |
| title_fullStr | Classical Potential Theory and Its Probabilistic Counterpart by J. L. Doob |
| title_full_unstemmed | Classical Potential Theory and Its Probabilistic Counterpart by J. L. Doob |
| title_short | Classical Potential Theory and Its Probabilistic Counterpart |
| title_sort | classical potential theory and its probabilistic counterpart |
| topic | Mathematics Potential theory (Mathematics) Distribution (Probability theory) Potential Theory Probability Theory and Stochastic Processes Mathematik Potenzialtheorie (DE-588)4046939-6 gnd Stochastischer Prozess (DE-588)4057630-9 gnd Martingal (DE-588)4126466-6 gnd Martingaltheorie (DE-588)4168982-3 gnd Wahrscheinlichkeitsrechnung (DE-588)4064324-4 gnd |
| topic_facet | Mathematics Potential theory (Mathematics) Distribution (Probability theory) Potential Theory Probability Theory and Stochastic Processes Mathematik Potenzialtheorie Stochastischer Prozess Martingal Martingaltheorie Wahrscheinlichkeitsrechnung |
| url | https://doi.org/10.1007/978-1-4612-5208-5 |
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