UID:
almahu_9947363143502882
Format:
XV, 323 p.
,
online resource.
ISBN:
9783642620256
Series Statement:
Classics in Mathematics,
Content:
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.
Note:
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 -- 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 -- 5.3. Time changes: Q = [0, + ?) -- 5.4. Local times -- 5.5. Subordination and chain rule -- 5.6. Killing times -- 5.7. Feller’s Brownian 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 ? -- 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.
In:
Springer eBooks
Additional Edition:
Printed edition: ISBN 9783540606291
Language:
English
DOI:
10.1007/978-3-642-62025-6
URL:
http://dx.doi.org/10.1007/978-3-642-62025-6
