Itô’s excursion theory and its applications

Pitman, Jim; Yor, Marc

doi:10.1007/s11537-007-0661-z

Cited by 20 publications

(17 citation statements)

References 32 publications

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“…The present paper also illustrates Pitman and Yor's discussion (see [42] in this volume) of K. Itô's general theory of excursions for a Markov process.…”

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confidence: 66%

On the excursion theory for linear diffusions

Salminen

Vallois²,

Yor

2007

Jpn. J. Math.

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We present a number of important identities related to the excursion theory of linear diffusions. In particular, excursions straddling an independent exponential time are studied in detail. Letting the parameter of the exponential time tend to zero it is seen that these results connect to the corresponding results for excursions of stationary diffusions (in stationary state). We characterize also the laws of the diffusion prior and posterior to the last zero before the exponential time. It is proved using Krein's representations that, e.g., the law of the length of the excursion straddling an exponential time is infinitely divisible. As an illustration of the results we discuss Ornstein-Uhlenbeck processes.

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“…The present paper also illustrates Pitman and Yor's discussion (see [42] in this volume) of K. Itô's general theory of excursions for a Markov process.…”

supporting

confidence: 66%

On the excursion theory for linear diffusions

Salminen

Vallois²,

Yor

2007

Jpn. J. Math.

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“…Let us recall Itô's excursion theory ( [23] and [30]). See also the standard textbooks [22] and [35], as well as [33].…”

Section: Discussion From Excursion Theoretic Viewpointmentioning

confidence: 99%

On the Laws of First Hitting Times of Points for One-Dimensional Symmetric Stable Lévy Processes

Yano

Yor

2009

Lecture Notes in Mathematics

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“…zero crossing, of continuous paths in great detail. The concept of local time, introduced by P. Lévy in the context of Brownian motion and Ito's excursion theory for continuous paths, are pillars in this field, see [17,18] and references within. Here we use a heuristic approach, with a modification of the original process v(t), by introducing a cutoff ± : the starting point of the particle after each zero hitting, which is taken to zero at the end.…”

Section: Introductionmentioning

confidence: 99%

From the Area under the Bessel Excursion to Anomalous Diffusion of Cold Atoms

2014

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Lévy flights are random walks in which the probability distribution of the step sizes is fat-tailed. Lévy spatial diffusion has been observed for a collection of ultracold 87 Rb atoms and single 24 Mg þ ions in an optical lattice, a system which allows for a unique degree of control of the dynamics. Using the semiclassical theory of Sisyphus cooling, we formulate the problem as a coupled Lévy walk, with strong correlations between the length χ and duration τ of the excursions. Interestingly, the problem is related to the area under the Bessel and Brownian excursions. These are overdamped Langevin motions that start and end at the origin, constrained to remain positive, in the presence or absence of an external logarithmic potential, respectively. In the limit of a weak potential, i.e., shallow optical lattices, the celebrated Airy distribution describing the areal distribution of the Brownian excursion is found as a limiting case. Three distinct phases of the dynamics are investigated: normal diffusion, Lévy diffusion and, below a certain critical depth of the optical potential, x ∼ t 3=2 scaling, which is related to Richardson's diffusion from the field of turbulence. The main focus of the paper is the analytical calculation of the joint probability density function ψðχ; τÞ from a newly developed theory of the area under the Bessel excursion. The latter describes the spatiotemporal correlations in the problem and is the microscopic input needed to characterize the spatial diffusion of the atomic cloud. A modified Montroll-Weiss equation for the Fourier-Laplace transform of the density Pðx; tÞ is obtained, which depends on the statistics of velocity excursions and meanders. The meander, a random walk in velocity space, which starts at the origin and does not cross it, describes the last jump event χ in the sequence. In the anomalous phases, the statistics of meanders and excursions are essential for the calculation of the mean-square displacement, indicating that our correction to the Montroll-Weiss equation is crucial and pointing to the sensitivity of the transport on a single jump event. Our work provides general relations between the statistics of velocity excursions and meanders on the one hand and the diffusivity on the other, both for normal and anomalous processes.

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Itô’s excursion theory and its applications

Abstract: A presentation of Itô's excursion theory for general Markov processes is given, with several applications to Brownian motion and related processes.

Cited by 20 publications

References 32 publications

On the excursion theory for linear diffusions

On the excursion theory for linear diffusions

On the Laws of First Hitting Times of Points for One-Dimensional Symmetric Stable Lévy Processes

From the Area under the Bessel Excursion to Anomalous Diffusion of Cold Atoms

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