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140122 ||| eng |
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|a 9783642620256
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|a Itô, Kiyosi
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|a Diffusion Processes and their Sample Paths
|h Elektronische Ressource
|c by Kiyosi Itô, Henry P. Jr. McKean
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| 250 |
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|a 1st ed. 1996
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| 260 |
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|a Berlin, Heidelberg
|b Springer Berlin Heidelberg
|c 1996, 1996
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| 300 |
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|a XV, 323 p
|b online resource
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|a 7.4. Greenian domains and Green functions -- 7.5. Excessive functions -- 7.6. Application to the spectrum of ?/2 -- 7.7. Potentials and hitting probabilities -- 7.8. Newtonian capacities -- 7.9. Gauss’s quadratic form -- 7.10. Wiener’s test -- 7.11. Applications of Wiener’s test -- 7.12. Dirichlet problem -- 7.13. Neumann problem -- 7.14. Space-time Brownian motion -- 7.15. Spherical Brownian motion and skew products -- 7.16. Spinning -- 7.17. An individual ergodic theorem for the standard 2-dimensional BROWNian motion -- 7.18. Covering Brownian motions -- 7.19. Diffusions with Brownian hitting probabilities -- 7.20. Right-continuous paths -- 7.21. Riesz potentials -- 8. A general view of diffusion in several dimensions -- 8.1. Similar diffusions -- 8.2. G as differential operator -- 8.3. Time substitutions -- 8.4. Potentials -- 8.5. Boundaries -- 8.6. Elliptic operators -- 8.7. Feller’s little boundary and tail algebras -- List of notations
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|a Prerequisites -- 1. The standard BRownian motion -- 1.1. The standard random walk -- 1.2. Passage times for the standard random walk -- 1.3. Hin?in’s proof of the de Moivre-laplace limit theorem -- 1.4. The standard Brownian motion -- 1.5. P. Lévy’s construction -- 1.6. Strict Markov character -- 1.7. Passage times for the standard Brownian motion -- 1.8. Kolmogorov’s test and the law of the iterated logarithm -- 1.9. P. Lévy’s Hölder condition -- 1.10. Approximating the Brownian motion by a random walk -- 2. Brownian local times -- 2.1. The reflecting Brownian motion -- 2.2. P. Lévy’s local time -- 2.3. Elastic Brownian motion -- 2.4. t+ and down-crossings -- 2.5. T+ as Hausdorff-Besicovitch 1/2-dimensional measure -- 2.6. Kac’s formula for Brownian functionals -- 2.7. Bessel processes -- 2.8. Standard Brownian local time -- 2.9. BrowNian excursions -- 2.10. Application of the Bessel process to Brownian excursions -- 2.11. A time substitution --
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|a 5.3. Time changes: Q = [0, + ?) -- 5.4. Local times -- 5.5. Subordination and chain rule -- 5.6. Killing times -- 5.7. Feller’sBrownian motions -- 5.8. Ikeda’s example -- 5.9. Time substitutions must come from local time integrals -- 5.10. Shunts -- 5.11. Shunts with killing -- 5.12. Creation of mass -- 5.13. A parabolic equation -- 5.14. Explosions -- 5.15. A non-linear parabolic equation -- 6. Local and inverse local times -- 6.1. Local and inverse local times -- 6.2. Lévy measures -- 6.3. t and the intervals of [0, + ?) - ? -- 6.4. A counter example: t and the intervals of [0, + ?) - ? -- 6.5a t and downcrossings -- 6.5b t as Hausdorff measure -- 6.5c t as diffusion -- 6.5d Excursions -- 6.6. Dimension numbers -- 6.7. Comparison tests -- 6.8. An individual ergodic theorem -- 7. Brownian motion in several dimensions -- 7.1. Diffusion in several dimensions -- 7.2. The standard Brownian motion in several dimensions -- 7.3. Wandering out to ? --
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|a 3. The general 1-dimensional diffusion -- 3.1. Definition -- 3.2.Markov times -- 3.3. Matching numbers -- 3.4. Singular points -- 3.5. Decomposing the general diffusion into simple pieces -- 3.6. Green operators and the space D -- 3.7. Generators -- 3.8. Generators continued -- 3.9. Stopped diffusion -- 4. Generators -- 4.1. A general view -- 4.2. G as local differential operator: conservative non-singular case -- 4.3. G as local differential operator: general non-singular case -- 4.4. A second proof -- 4.5. G at an isolated singular point -- 4.6. Solving G•u = ? u -- 4.7. G as global differential operator: non-singular case -- 4.8. G on the shunts -- 4.9. G as global differential operator: singular case -- 4.10. Passage times -- 4.11. Eigen-differential expansions for Green functions and transition densities -- 4.12. Kolmogorov’s test -- 5. Time changes and killing -- 5.1. Construction of sample paths: a general view -- 5.2. Time changes: Q = R1 --
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| 653 |
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|a Probability Theory
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| 653 |
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|a Probabilities
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| 700 |
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|a McKean, Henry P. Jr
|e [author]
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| 041 |
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|a eng
|2 ISO 639-2
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| 989 |
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|b SBA
|a Springer Book Archives -2004
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| 490 |
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|a Classics in Mathematics
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| 028 |
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|a 10.1007/978-3-642-62025-6
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| 856 |
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|u https://doi.org/10.1007/978-3-642-62025-6?nosfx=y
|x Verlag
|3 Volltext
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| 082 |
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|a 519.2
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| 520 |
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|a Since its first publication in 1965 in the series Grundlehren der mathematischen Wissenschaften this book has had a profound and enduring influence on research into the stochastic processes associated with diffusion phenomena. Generations of mathematicians have appreciated the clarity of the descriptions given of one- or more- dimensional diffusion processes and the mathematical insight provided into Brownian motion. Now, with its republication in the Classics in Mathematics it is hoped that a new generation will be able to enjoy the classic text of Itô and McKean
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