Diffusion Processes and Their Sample Paths.

Abstract

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. Levy's construction.- 1.6. Strict Markov character.- 1.7. Passage times for the standard Brownian motion.- Note l: Homogeneous differential processes with increasing paths.- 1.8. Kolmogorov's test and the law of the iterated logarithm.- 1.9. P. Levy's Holder condition.- 1.10. Approximating the Brownian motion by a random walk.- 2. Brownian local times.- 2.1. The reflecting Brownian motion.- 2.2. P. Levy's local time.- 2.3. Elastic Brownian motion.- 2.4. t+ and down-crossings.- 2.5. T+ as Hausdorff-Besicovitch 1/2-dimensional measure.- Note 1: Submartingales.- Note 2: Hausdorff measure and dimension.- 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.- Note 1: Differential processes with increasing paths.- 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. Levy 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.- Note 1: Dimension numbers and fractional dimensional capacities.- 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.

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