擴散過程及其樣本軌道

內容介紹

《擴散過程及其樣本軌道(英文版)》是Springer《數學經典教材》系列之一，對與擴散現象有關的隨機過程產生持久而深刻的影響。不少數學家受益於《擴散過程及其樣本軌道(英文版)》一維和多維擴散過程的描述和獨到的布朗運動數學見解。傳承這一系列書的風格，行文簡潔流暢。每章節後面都配有問題並有部分解答，很適合作為教材和自學用書。目次：標準布朗運動；布朗局部時間；一般一維擴散；生成元；局部時間和逆時間序列；多維布朗運動；多維擴散簡述。

作品目錄

PrerequisitesChapter 1. The standard BRowNian motion1.1.The standard random walk1.2.Passage times for the standard random walk1.3.HINCIN'S proof of the DE MOIVRE-LAPLACE limit theorem1.4.The standard BROWNian motion1.5.P. LEVY's construction1.6.Strict MAgKOV character1.7.Passage times for the standard BgowNian motion Note 1: Homogeneous differential processes with increasing paths1.8.KOLMOGOROV'S test and the law of the iterated logarithm1.9.P. LEVY'S HOLDER condition1.10. Approximating the BgowNian motion by a random walkChapter 2. BROWNian local times2.1.The reflecting BRowNian motion2.2.P. LEVY'S local time2.3.Elastic BgowNian motion2.4.t+ and down-crossings2.5.t+ as HAUSDORFF-BESICOVITCH 1/2-dimensional measure Note 1: Submartingales Note 2: HAUSDORFF measure and dimension2.6.Kxc's formula for BRowNian funetionals2.7.BESSEL processes2.8.Standard BRowNian local time2.9.BRowNian excursions2.10. Application of the BESSEL process to BROWNian excursions2.11. A time substitutionChapter 3. The general t-dimensional diffusion3.t.Definition3.2.MARKOV times3.3.Matching numbers3.4.Singular points3.5.Decomposing the general diffusion into simple pieces3.6.GREEN operators and the spaceD3.7.Generators3.8.Generators continued3.9.Stopped diffusionChapter 4. Generators 4.1.A general view4.2. as local differential operator: conservative non-singular case4.3. as local differential operator: general non-singular case4.4.A second proof4.5. at an isolated singular point4.6.Solving4.7. as global differential operator: non-singular case4.8. on the shunts4.9. as global differential operator: singular case4.10. Passage times Note 1: Differential processes with increasing paths4.ft. Eigen-differential expansions for GREEN functions and transition densities4.12. KOLMOGOROV'S testChapter 5. Time changes and killing5.1.Construction of sample paths: a general view5.2.Time changes5.3.Time changes5.4.Local times5.5.Subordination and chain rule5.6.Killing times5.7.FELLER'S BROWNlan motions5.8.IKEDA'S example5.9.Time substitutions must come from local time integrals5.10. Shunts5.11. Shunts with killing5.12. Creation of mass5.13. A parabolic equation5.f4. Explosions5.15. A non-linear parabolic equationChapter 6. Local and inverse local times6.1.Local and inverse local times6.2.LEVY measures6.3.t and the intervals of [0, + ∞)6.4.A counter example: t and the intervals of [0, +∞)6.5a t and downcrossings6.5b t as HAUSDORFF measure6.5c t as diffusion6.5d Excursions6.6.Dimension numbers6.7.Comparison tests Notension Dimension numbers and fractional dimensional capacities6.8.An individual ergodic theoremChapter 7. BRowNian motion in several dimensions7.1.Diffusion in several dimensions7.2.The standard BRowNian motion in several dimensions7.3.Wandering out to oo7.4.GREENian domains and GREEN functions7.5.Excessive functions7.6.Application to the spectrum of /1/27.7.Potentials and hitting probabilities7.8.NEWTONian capacities7.9.GAUSS's quadratic form7.10. WIENER'S test7.11. Applicatiors of WIENER'S test7.12. DIRICHLET problem7.13. NEUHANN problem7.14. Space-time BROWNian moticn7.15. Spherical BROWNian motion and skew products7.16. Spinning7.17. An individualergodic theorem for the standard 2-dimensional BROWNian motion7.18. Covering BROWNian motions7.19. Diffusions with BROWNian hitting probabilities7.20. Right-continuous paths7.21. RIESZ potentialsChapter 8. A general view of diffusion in several dimensions8.1.Similar diffusions8.2.as differential operater8.3.Time substitutions8.4.Potentials8.5.Boundaries8.6.Elliptic operators8.7.FELLER'S little boundary and tail algebrasBibliographyList of notationsIndex