Some Aspects of Brownian Motion

1. Front Matter

   Pages I-XII

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2. On principal values of Brownian and Bessel local times

   • Marc Yor

   Pages 1-10

3. Probabilistic representations of the Riemann zeta function and some generalisations related to Bessel processes

   • Marc Yor

   Pages 11-31

4. Some examples and applications of enlargements of filtrations

   • Marc Yor

   Pages 32-50

5. Martingale inequalities at any time

   • Marc Yor

   Pages 51-60

6. On the martingales which vanish on the set of Brownian zeroes

   • Marc Yor

   Pages 61-78

7. On Azéma’s martingales and the chaos representation property

   • Marc Yor

   Pages 79-93

8. The filtration of truncated Brownian motion

   • Marc Yor

   Pages 94-102

9. The Brownian filtration, Tsirel’son’s examples, and Walsh’s Brownian motions

   • Marc Yor

   Pages 103-123

10. Complements relative to Part I (Chapters 1 to 9)

    • Marc Yor

    Pages 124-134

11. Back Matter

    Pages 135-148

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The following notes represent approximately the second half of the lectures I gave in the Nachdiplomvorlesung, in ETH, Zurich, between October 1991 and February 1992, together with the contents of six additional lectures I gave in ETH, in November and December 1993. Part I, the elder brother of the present book [Part II], aimed at the computation, as explicitly as possible, of a number of interesting functionals of Brownian motion. It may be natural that Part II, the younger brother, looks more into the main technique with which Part I was "working", namely: martingales and stochastic calculus. As F. Knight writes, in a review article on Part I, in which research on Brownian motion is compared to gold mining: "In the days of P. Levy, and even as late as the theorems of "Ray and Knight" (1963), it was possible for the practiced eye to pick up valuable reward without the aid of much technology . . . Thereafter, however, the rewards are increasingly achieved by the application of high technology". Although one might argue whether this golden age is really foregone, and discuss the "height" of the technology involved, this quotation is closely related to the main motivations of Part II: this technology, which includes stochastic calculus for general discontinuous semi-martingales, enlargement of filtrations, . . .

• Brownian motion
• Markov process
• Martingale
• Probability theory
• filtration
• function
• local time

"This is a formidable book, and is meant for the specialist...this book (together with Part I) can be very rewarding, imparting a very good insight into the subtelities of martinglae theory and stochastic calculus."   

--The Journal of the Indian Inst. of Science

 

"All efforts are rewarded by acknowledgment with modern point of view on such a popular object as Brownian motion and on still open problems related to it." -Zeitschrift fur Mathematik

• Laboratoire de Probabilités, Université Pierre et Màrie Curie, Paris Cedex 05, France

  Marc Yor

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