Results 1  10
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14
Rates of convergence of diffusions with drifted Brownian potentials
 Trans. Amer. Math. Soc
, 1999
"... Abstract. 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 d ..."
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Cited by 13 (5 self)
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Abstract. 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. 1.
A Vervaatlike path transformation for the reflected Brownian bridge conditioned on its local time at 0
, 1999
"... We describe a Vervaatlike path transformation for the reflected Brownian bridge conditioned on its local time at 0: up to random shifts, this process equals the two processes constructed from a Brownian bridge and a Brownian excursion by adding a drift and then taking the excursions over the cur ..."
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Cited by 10 (1 self)
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We describe a Vervaatlike path transformation for the reflected Brownian bridge conditioned on its local time at 0: up to random shifts, this process equals the two processes constructed from a Brownian bridge and a Brownian excursion by adding a drift and then taking the excursions over the current minimum. As a consequence, these three processes have the same occupation measure, which is easily found. The three processes arise as limits, in three different ways, of profiles associated to hashing with linear probing, or, equivalently, to parking functions. 1 Introduction We regard the Brownian bridge b(t) and the normalized (positive) Brownian excursion e(t) as defined on the circle R=Z, or, equivalently, as defined on the whole real line, being periodic with period 1. We define, for a 0, the operator \Psi a on the set of bounded functions on the line by \Psi a f(t) = f(t) \Gamma at \Gamma inf \Gamma1!st (f(s) \Gamma as) = sup st (f(t) \Gamma f(s) \Gamma a(t \Gamma s))...
Some Asymptotic Properties of the Local Time of the Uniform Empirical Process
, 1998
"... this paper we are interested in strong limit theorems for the local time of ff n . We first recall two important results. For notational convenience, we write ..."
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Cited by 6 (0 self)
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this paper we are interested in strong limit theorems for the local time of ff n . We first recall two important results. For notational convenience, we write
Large Favourite Sites Of Simple Random Walk And The Wiener Process
 Electronic J. Probab
, 1998
"... Let U(n) denote the most visited point by a simple symmetric random walk fS k g k0 in the first n steps. It is known that U(n) and max 0kn S k satisfy the same law of the iterated logarithm, but have different upper functions (in the sense of P. Lévy). The distance between them however turns out to ..."
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Cited by 2 (1 self)
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Let U(n) denote the most visited point by a simple symmetric random walk fS k g k0 in the first n steps. It is known that U(n) and max 0kn S k satisfy the same law of the iterated logarithm, but have different upper functions (in the sense of P. Lévy). The distance between them however turns out to be transient. In this paper, we establish the exact rate of escape of this distance. The corresponding problem for the Wiener process is also studied.
Hitting, Occupation, and Inverse Local Times of OneDimensional Diffusions: Martingale and Excursion Approaches
, 2001
"... Basic relations between the distributions of hitting, occupation, and inverse local times of a onedimensional diffusion process X , first discussed by ItoMcKean, are reviewed from the perspectives of martingale calculus and excursion theory. These relations, and the technique of conditioning o ..."
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Cited by 2 (1 self)
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Basic relations between the distributions of hitting, occupation, and inverse local times of a onedimensional diffusion process X , first discussed by ItoMcKean, are reviewed from the perspectives of martingale calculus and excursion theory. These relations, and the technique of conditioning on L y T , the local time of X at level y before a suitable random time T , yield formulae for the joint Laplace transform of L y T and the times spent by X above and below level y up to time T . Contents 1
Constructions Of A Brownian Pathwith A Given Minimum
, 1999
"... We construct a Brownian path conditioned on its minimum value over a fixed time interval by simple transformations of a Brownian bridge. Path transformations have proved useful in the study of Brownian motion and related processes, by providing simple constructions of various conditioned processes ..."
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Cited by 2 (0 self)
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We construct a Brownian path conditioned on its minimum value over a fixed time interval by simple transformations of a Brownian bridge. Path transformations have proved useful in the study of Brownian motion and related processes, by providing simple constructions of various conditioned processes such as Brownian bridge, meander and excursion, starting from an unconditioned Brownian motion. As well as providing insight into the structure of these conditioned processes, path constructions assist in the computation of various conditional laws of Brownian functionals, and in the simulation of conditioned processes. Starting from a standard onedimensional Brownian motion B =(B t ) 0#t#1 with B 0 =0, one well known construction of a Brownian bridge of length 1 from 0 to x, denoted B br,x ,is the following: B br,x u := B u  uB 1 + ux (0 # u # 1). (1) Then a Brownian meander of length 1 starting at 0 and conditioned to end at r # 0, denoted B me,r , can be constructed from...
Simulation of a stochastic process in a discontinuous, layered media
, 2011
"... In this note, we provide a simulation algorithm for a diffusion process in a layered media. Our main tools are the properties of the Skew Brownian motion and a path decomposition technique for simulating occupation times. ..."
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Cited by 2 (1 self)
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In this note, we provide a simulation algorithm for a diffusion process in a layered media. Our main tools are the properties of the Skew Brownian motion and a path decomposition technique for simulating occupation times.
A double phase transition arising from Brownian entropic repulsion
, 2008
"... Abstract. We analyze onedimensional Brownian motion conditioned on a selfrepelling behaviour. In the main result of this paper, it is shown that a double phase transition occurs when the growth of the local time at the origin is constrained (in a suitable way) to be slower than the function f(t) ..."
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Cited by 1 (1 self)
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Abstract. We analyze onedimensional Brownian motion conditioned on a selfrepelling behaviour. In the main result of this paper, it is shown that a double phase transition occurs when the growth of the local time at the origin is constrained (in a suitable way) to be slower than the function f(t) = √ t(log t) −c at every time. In the subcritical phase (c < 0), the process is recurrent and the local time at 0 is diffusive. In the intermediary phase (0 < c ≤ 1), the process is recurrent but the local time grows much slower than the constraint f. Finally in the supercritical phase (c> 1), the process becomes transient. The proof exploits the Brownian entropic repulsion phenomenon.