Results 1  10
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14
Rates of convergence of diffusions with drifted Brownian potentials
 TRANS. AMER. MATH. SOC
, 1999
"... We are interested in the asymptotic behaviour of a diffusion process with drifted Brownian potential. The model is a continuous time analogue to the random walk in random environment studied in the classical paper of Kesten, Kozlov, and Spitzer. We not only recover the convergence of the diffusion ..."
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Cited by 20 (5 self)
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We are interested in the asymptotic behaviour of a diffusion process with drifted Brownian potential. The model is a continuous time analogue to the random walk in random environment studied in the classical paper of Kesten, Kozlov, and Spitzer. We not only recover the convergence of the diffusion process which was previously established by Kawazu and Tanaka, but also obtain all the possible convergence rates. An interesting feature of our approach is that it shows a clear relationship between drifted Brownian potentials and Bessel processes.
The Problem of the Most Visited Site in Random Environment
, 2000
"... this paper is to study the favourite points. For n 0 and x 2 Z, ..."
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Cited by 11 (1 self)
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this paper is to study the favourite points. For n 0 and x 2 Z,
RATES OF CONVERGENCE OF A TRANSIENT DIFFUSION IN A SPECTRALLY NEGATIVE LÉVY POTENTIAL
, 2008
"... We consider a diffusion process X in a random Lévy potential V which is a solution of the informal stochastic differential equation dXt = dβt − ..."
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Cited by 9 (1 self)
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We consider a diffusion process X in a random Lévy potential V which is a solution of the informal stochastic differential equation dXt = dβt −
Lyapunov exponents, onedimensional Anderson localisation and products of random matrices
 J. Phys. A: Math. Theor
, 2013
"... Abstract. The concept of Lyapunov exponent has long occupied a central place in the theory of Anderson localisation; its interest in this particular context is that it provides a reasonable measure of the localisation length. The Lyapunov exponent also features prominently in the theory of products ..."
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Cited by 3 (2 self)
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Abstract. The concept of Lyapunov exponent has long occupied a central place in the theory of Anderson localisation; its interest in this particular context is that it provides a reasonable measure of the localisation length. The Lyapunov exponent also features prominently in the theory of products of random matrices pioneered by Furstenberg. After a brief historical survey, we describe some recent work that exploits the close connections between these topics. We review the known solvable cases of disordered quantum mechanics involving random point scatterers and discuss a new solvable case. Finally, we point out some limitations of the Lyapunov exponent as a means of studying localisation properties.
The maximum of the local time of a diffusion process in a drifted Brownian potential
, 2006
"... ..."
Devulder A.: Localization and number of visited valleys for a transient diffusion in random environment. Preprint ArXiv
, 2013
"... Abstract. We consider a transient diffusion in a (−κ/2)drifted Brownian potential Wκ with 0 < κ < 1. We prove its localization before time t in an a neighborhood of some random points depending only on the environment, which are the positive htminima of the environment, for ht a bit smaller ..."
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Cited by 1 (0 self)
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Abstract. We consider a transient diffusion in a (−κ/2)drifted Brownian potential Wκ with 0 < κ < 1. We prove its localization before time t in an a neighborhood of some random points depending only on the environment, which are the positive htminima of the environment, for ht a bit smaller than log t. We also prove an Aging phenomenon for the diffusion, and provide a central limit theorem for the number of valleys visited up to time t. The proof relies on a Williams ’ decomposition of the trajectory ofWκ in the neighborhood of local minima, with the help of results of Faggionato [19], and on a precise analysis of exponential functionals of Wκ and of 3dimensional (−κ/2)drifted Bessel processes. 1. Introduction and
Annealed tail estimates for a Brownian motion in a drifted Brownian potential
, 2006
"... We study Brownian motion in a drifted Brownian potential. Kawazu and Tanaka [23] exhibited two speed regimes for this process, depending on the drift. They supplemented these laws of large numbers by central limit theorems, which were recently completed by Hu, Shi and Yor [19] using stochastic cal ..."
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We study Brownian motion in a drifted Brownian potential. Kawazu and Tanaka [23] exhibited two speed regimes for this process, depending on the drift. They supplemented these laws of large numbers by central limit theorems, which were recently completed by Hu, Shi and Yor [19] using stochastic calculus. We studied large deviations [34], showing among other results that the rate function in the annealed setting, that is after averaging over the potential, has a flat piece in the ballistic regime. In this paper, we focus on this subexponential regime, proving that the probability of deviating below the almost sure speed has a polynomial rate of decay, and computing the exponent in this power law. This provides the continuoustime analogue of what Dembo, Peres and Zeitouni proved for the transient random walk in random environment [13]. Our method takes a completely different route, making use of Lamperti’s representation together with an iteration scheme.
LOCAL TIME OF A DIFFUSION IN A STABLE LÉVY ENVIRONMENT
, 2009
"... We consider a onedimensional diffusion in a stable Lévy environment. We show that the normalized local time process refocused at the bottom of the standard valley with height log t, (LX(t,mlog t + x)/t,x ∈ R), converges in law to a functional of two independent Lévy processes conditioned to stay p ..."
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We consider a onedimensional diffusion in a stable Lévy environment. We show that the normalized local time process refocused at the bottom of the standard valley with height log t, (LX(t,mlog t + x)/t,x ∈ R), converges in law to a functional of two independent Lévy processes conditioned to stay positive. To prove this result, we show that the law of the standard valley is close to a twosided Lévy process conditioned to stay positive. We also obtain the limit law of the supremum of the normalized local time. This result has been obtained by Andreoletti and Diel [1] in the case of a Brownian environment.