Les temps de passage successifs de l'intégrale du mouvement brownien

Lachal, Aimé

doi:10.1016/s0246-0203(97)80114-8

Cited by 20 publications

(22 citation statements)

References 10 publications

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“…The statistics of the times of first passage by the origin, and of related quantities, for a particle obeying (1), has been the subject of a long series of works, and is by now well understood [1][2][3][4][5].…”

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confidence: 99%

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Partial survival and inelastic collapse for a randomly accelerated particle

Smedt¹,

Godrèche²,

Luck³

2001

Europhys. Lett.

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Abstract. -We present an exact derivation of the survival probability of a randomly accelerated particle subject to partial absorption at the origin. We determine the persistence exponent and the amplitude associated to the decay of the survival probability at large times. For the problem of inelastic reflection at the origin, with coefficient of restitution r, we give a new derivation of the condition for inelastic collapse, r < rc = e −π/ √ 3 , and determine the persistence exponent exactly. Consider a randomly accelerated particle, obeying the stochastic equation of motionwhere η t is Gaussian white noise with zero average, η t = 0, and correlator η t η t ′ = δ(t − t ′ ). This is the original Langevin equation with no damping force. The joint random variables (x t , v t ) evolve in time aswith initial conditions (x 0 , v 0 ). Thereforewhere W t is the integral of the noise, or Brownian motion, hence the process x t is usually referred to as the integral of Brownian motion. The statistics of the times of first passage by the origin, and of related quantities, for a particle obeying (1), has been the subject of a long series of works, and is by now well understood [1][2][3][4][5].More recently, a number of studies have been devoted to survival problems for a randomly accelerated particle, with particular choices of the boundary conditions at the origin, motivated by situations of physical interest.c EDP Sciences

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confidence: 99%

“…We shall also give an analytical prediction for the amplitude A(p) of the powerlaw (2) [10]. We use the framework of ideas and results contained in the classical work of McKean [1], complemented by subsequent studies by Lachal [5].…”

mentioning

confidence: 99%

Partial survival and inelastic collapse for a randomly accelerated particle

Smedt¹,

Godrèche²,

Luck³

2001

Europhys. Lett.

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“…This construction requires to estimate the excursion times at 0 of the primitive of the Brownian motion. In [18,20], the authors prove that, starting from x 0 > 0, the 1D-Brownian's primitive has only countably many excursions from 0, they also explicit the law of the related hitting sequence.…”

Section: Confinement In An Hyperplanementioning

confidence: 99%

Stochastic Lagrangian method for downscaling problems in computational fluid dynamics

Bernardin¹,

Bossy²,

Chauvin³

et al. 2010

ESAIM: M2AN

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Abstract. This work aims at introducing modelling, theoretical and numerical studies related to a new downscaling technique applied to computational fluid dynamics. Our method consists in building a local model, forced by large scale information computed thanks to a classical numerical weather predictor. The local model, compatible with the Navier-Stokes equations, is used for the small scale computation (downscaling) of the considered fluid. It is inspired by Pope's works on turbulence, and consists in a so-called Langevin system of stochastic differential equations. We introduce this model and exhibit its links with classical RANS models. Well-posedness, as well as mean-field interacting particle approximations and boundary condition issues are addressed. We present the numerical discretization of the stochastic downscaling method and investigate the accuracy of the proposed algorithm on simplified situations.Mathematics Subject Classification. 65C20, 65C35, 68U20, 35Q30.

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“…For the simplest double integral process (DIP), the integrated Wiener process (IWP) defined in (5), below, McKean [18], Goldman [10], and Lachal [14], [15], [16] found the probability distribution of the first hitting time to a constant boundary using stochastic calculus methods. Lefebvre [17] used the Kolmogorov (Fokker-Planck) equation to find in some special cases closed-form solutions.…”

Section: J Touboul and O Faugerasmentioning

confidence: 99%

A characterization of the first hitting time of double integral processes to curved boundaries

Touboul

Faugeras

2008

Adv. Appl. Probab.

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The problem of finding the probability distribution of the first hitting time of a double integral process (DIP) such as the integrated Wiener process (IWP) has been an important and difficult endeavor in stochastic calculus. It has applications in many fields of physics (first exit time of a particle in a noisy force field) or in biology and neuroscience (spike time distribution of an integrate-and-fire neuron with exponentially decaying synaptic current). The only results available are an approximation of the stationary mean crossing time and the distribution of the first hitting time of the IWP to a constant boundary. We generalize these results and find an analytical formula for the first hitting time of the IWP to a continuous piecewise-cubic boundary. We use this formula to approximate the law of the first hitting time of a general DIP to a smooth curved boundary, and we provide an estimation of the convergence of this method. The accuracy of the approximation is computed in the general case for the IWP and the effective calculation of the crossing probability can be carried out through a Monte Carlo method.

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Les temps de passage successifs de l'intégrale du mouvement brownien

Cited by 20 publications

References 10 publications

Partial survival and inelastic collapse for a randomly accelerated particle

Partial survival and inelastic collapse for a randomly accelerated particle

Stochastic Lagrangian method for downscaling problems in computational fluid dynamics

A characterization of the first hitting time of double integral processes to curved boundaries

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