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基本説明
伊藤清・京大名誉教授は、国際数学連合が数学の応用で社会に大きな影響を与えた学者に贈る「ガウス賞」の、第１回受賞者（2006年8月22日受賞）。
Table of contents : The standard Brownian motion. Brownian local times. The general 1-dimensional diffusion. Generators. Time changes and killing. Local and inverse local times. Browinian motion in several dimensions. A general view of diffusion in several dimentions. Bibliography. List of notations. Index.
Full Description
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.
Contents
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'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 ?.- 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.
