Diffusions and Elliptic Operators:
Gespeichert in:
| Beteilige Person: | |
|---|---|
| Format: | Elektronisch E-Book |
| Sprache: | Englisch |
| Veröffentlicht: |
New York, NY
Springer New York
1998
|
| Schriftenreihe: | Probability and its Applications, A Series of the Applied Probability Trust
|
| Schlagwörter: | |
| Links: | https://doi.org/10.1007/b97611 |
| Beschreibung: | The interplay of probability theory and partial differential equations forms a fascinating part of mathematics. Among the subjects it has inspired are the martingale problems of Stroock and Varadhan, the Harnack inequality of Krylov and Safonov, the theory of symmetric diffusion processes, and the Malliavin calculus. When I first made an outline for my previous book Probabilistic Techniques in Analysis, I planned to devote a chapter to these topics. I soon realized that a single chapter would not do the subject justice, and the current book is the result. The first chapter provides the probabilistic machine needed to drive the subject, namely, stochastic differential equations. We consider existence, uniqueness, and smoothness of solutions and stochastic differential equations with reflection. The second chapter is the heart of the subject. We show how many partial differential equations can be solved by simple probabilistic expressions. The Dirichlet problem, the Cauchy problem, the Neumann problem, the oblique derivative problem, Poisson's equation, and Schrödinger's equation all have solutions that are given by appropriate probabilistic expressions. Green functions and fundamental solutions also have simple probabilistic representations. If an operator has smooth coefficients, then equations with these operators will have smooth solutions. This theory is discussed in Chapter III. The chapter is largely analytic, but probability allows some simplification in the arguments. Chapter IV considers one-dimensional diffusions and the corresponding second-order ordinary differential equations. Every one-dimensional diffusion viii PREFACE can be derived from Brownian motion by changes of time and scale |
| Umfang: | 1 Online-Ressource (XIV, 232 p) |
| ISBN: | 9780387226040 9780387983158 |
| ISSN: | 1431-7028 |
| DOI: | 10.1007/b97611 |
Internformat
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| 500 | |a The interplay of probability theory and partial differential equations forms a fascinating part of mathematics. Among the subjects it has inspired are the martingale problems of Stroock and Varadhan, the Harnack inequality of Krylov and Safonov, the theory of symmetric diffusion processes, and the Malliavin calculus. When I first made an outline for my previous book Probabilistic Techniques in Analysis, I planned to devote a chapter to these topics. I soon realized that a single chapter would not do the subject justice, and the current book is the result. The first chapter provides the probabilistic machine needed to drive the subject, namely, stochastic differential equations. We consider existence, uniqueness, and smoothness of solutions and stochastic differential equations with reflection. The second chapter is the heart of the subject. We show how many partial differential equations can be solved by simple probabilistic expressions. The Dirichlet problem, the Cauchy problem, the Neumann problem, the oblique derivative problem, Poisson's equation, and Schrödinger's equation all have solutions that are given by appropriate probabilistic expressions. Green functions and fundamental solutions also have simple probabilistic representations. If an operator has smooth coefficients, then equations with these operators will have smooth solutions. This theory is discussed in Chapter III. The chapter is largely analytic, but probability allows some simplification in the arguments. Chapter IV considers one-dimensional diffusions and the corresponding second-order ordinary differential equations. Every one-dimensional diffusion viii PREFACE can be derived from Brownian motion by changes of time and scale | ||
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Datensatz im Suchindex
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| discipline | Mathematik |
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| id | DE-604.BV042419042 |
| illustrated | Not Illustrated |
| indexdate | 2024-12-20T17:10:39Z |
| institution | BVB |
| isbn | 9780387226040 9780387983158 |
| issn | 1431-7028 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-027854459 |
| oclc_num | 704424515 |
| open_access_boolean | |
| owner | DE-384 DE-703 DE-91 DE-BY-TUM DE-634 |
| owner_facet | DE-384 DE-703 DE-91 DE-BY-TUM DE-634 |
| physical | 1 Online-Ressource (XIV, 232 p) |
| psigel | ZDB-2-SMA ZDB-2-BAE ZDB-2-SMA_Archive |
| publishDate | 1998 |
| publishDateSearch | 1998 |
| publishDateSort | 1998 |
| publisher | Springer New York |
| record_format | marc |
| series2 | Probability and its Applications, A Series of the Applied Probability Trust |
| spellingShingle | Bass, Richard F. 1951- Diffusions and Elliptic Operators Mathematics Global analysis (Mathematics) Distribution (Probability theory) Probability Theory and Stochastic Processes Analysis Mathematik Elliptischer Differentialoperator (DE-588)4140057-4 gnd Diffusion (DE-588)4012277-3 gnd Diffusionsprozess (DE-588)4274463-5 gnd Stochastische Differentialgleichung (DE-588)4057621-8 gnd |
| subject_GND | (DE-588)4140057-4 (DE-588)4012277-3 (DE-588)4274463-5 (DE-588)4057621-8 |
| title | Diffusions and Elliptic Operators |
| title_auth | Diffusions and Elliptic Operators |
| title_exact_search | Diffusions and Elliptic Operators |
| title_full | Diffusions and Elliptic Operators by Richard F. Bass |
| title_fullStr | Diffusions and Elliptic Operators by Richard F. Bass |
| title_full_unstemmed | Diffusions and Elliptic Operators by Richard F. Bass |
| title_short | Diffusions and Elliptic Operators |
| title_sort | diffusions and elliptic operators |
| topic | Mathematics Global analysis (Mathematics) Distribution (Probability theory) Probability Theory and Stochastic Processes Analysis Mathematik Elliptischer Differentialoperator (DE-588)4140057-4 gnd Diffusion (DE-588)4012277-3 gnd Diffusionsprozess (DE-588)4274463-5 gnd Stochastische Differentialgleichung (DE-588)4057621-8 gnd |
| topic_facet | Mathematics Global analysis (Mathematics) Distribution (Probability theory) Probability Theory and Stochastic Processes Analysis Mathematik Elliptischer Differentialoperator Diffusion Diffusionsprozess Stochastische Differentialgleichung |
| url | https://doi.org/10.1007/b97611 |
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