Titlebook: Some Aspects of Brownian Motion; Part II: Some Recent Marc Yor Book 1997 Springer Basel AG 1997 Brownian motion.Markov process.Martingale.P

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Book 1997logy . . . 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

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uch 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 II978-3-7643-5717-7978-3-0348-8954-4

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adduction

On principal values of Brownian and Bessel local times,artly because of the fundamental identity between Fourier transforms:.If, in (10.1), . is assumed to be H?lder continuous, and has compact support, then the limit in . exists for every..This remark applies to., the function, in the space variable y, of the local times of Brownian motion at time .

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On the martingales which vanish on the set of Brownian zeroes,alue 0) on the set.≡{(t,w):.(w)=0}of the zeroes of a one-dimensional .≥0). Such martingales were also encountered naturally in relation with the balayage formula (see, e.g., Azéma-Yor [88]). It thus seems interesting to try and describe the class.as well as

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Probabilistic representations of the Riemann zeta function and some generalisations related to BessTo begin with, it may be wise to state immediately that the aim of this chapter is not to discuss Riemann’s hypothesis!, but, much more modestly, to present some of the (well-known) relations between heat equation, zeta function, theta functions and Brownian motion.

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Some examples and applications of enlargements of filtrations,It?’s stochastic integration provides the definition of integrals.of processes ? which are “non-anticipating” with respect to the filtration.of the past of the Brownian motion (B., . ≥ 0).

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Martingale inequalities at any time,If (M., . ≥ 0) is any continuous local martingale, such that M., = 0, the inequalities of Burkholder-Gundy . inequalities, in the sequel) tell us that, for any . 0, there exist two universal constants 0 < c.< C. .∞ such that:.where.. and.

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