Results 1  10
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36
On some exponential functionals of Brownian motion
 Adv. Appl. Prob
, 1992
"... Abstract: This is the second part of our survey on exponential functionals of Brownian motion. We focus on the applications of the results about the distributions of the exponential functionals, which have been discussed in the first part. Pricing formula for call options for the Asian options, expl ..."
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Cited by 119 (12 self)
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Abstract: This is the second part of our survey on exponential functionals of Brownian motion. We focus on the applications of the results about the distributions of the exponential functionals, which have been discussed in the first part. Pricing formula for call options for the Asian options, explicit expressions for the heat kernels on hyperbolic spaces, diffusion processes in random environments and extensions of Lévy’s and Pitman’s theorems are discussed.
Random walks in random environment
 In Lectures on probability theory and statistics
, 2004
"... Random walks in random environments (RWRE’s) have been a source of surprising phenomena and challenging problems since they began to be studied in the 70’s. Hitting times and, more recently, certain regeneration structures, have played a major role in our understanding of RWRE’s. We review these and ..."
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Cited by 70 (11 self)
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Random walks in random environments (RWRE’s) have been a source of surprising phenomena and challenging problems since they began to be studied in the 70’s. Hitting times and, more recently, certain regeneration structures, have played a major role in our understanding of RWRE’s. We review these and provide some hints on current research directions and challenges.
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 15 (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 9 (1 self)
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this paper is to study the favourite points. For n 0 and x 2 Z,
The Mean Velocity Of A Brownian Motion In A Random Lévy Potential
"... . A Brownian motion in a random L'evy potential V , is the informal solution of the stochastic differential equation dX t = dB t \Gamma 1 2 V 0 (X t ) dt ; where B is a Brownian motion independent of V . We generalize some results of KawazuTanaka [8], who considered for V a Brownian moti ..."
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Cited by 8 (0 self)
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. A Brownian motion in a random L'evy potential V , is the informal solution of the stochastic differential equation dX t = dB t \Gamma 1 2 V 0 (X t ) dt ; where B is a Brownian motion independent of V . We generalize some results of KawazuTanaka [8], who considered for V a Brownian motion with drift, by proving that X t t converges almost surely to a constant, the mean velocity, which we compute in terms of the L'evy exponent OE of V , defined by : E \Theta e mV (t) = e \GammatOE(m) . 1. Introduction Example 1. Given a Poisson cloud (oe i ; i 2 Z) on the real line, we consider a process X such that: ffl X behaves like a Brownian motion between adjacent barriers oe i and oe i+1 ; ffl when X hits a barrier, he flips a coin and goes to the right with probability p, to the left with probability q = 1 \Gamma p. Observe that it is natural to require stationarity of the random media, that is invariance in law under translations, so that the intervals Date: November 2...
On the concentration of Sinai’s walk
, 2008
"... Abstract: We consider Sinai’s random walk in random environment. We prove that for an interval of time [1,n] Sinai’s walk sojourns in a small neighborhood of the point of localization for the quasi totality of this amount of time. Moreover the local time at the point of localization normalized by n ..."
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Cited by 7 (7 self)
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Abstract: We consider Sinai’s random walk in random environment. We prove that for an interval of time [1,n] Sinai’s walk sojourns in a small neighborhood of the point of localization for the quasi totality of this amount of time. Moreover the local time at the point of localization normalized by n converges in probability to a well defined random variable of the environment. From these results we get applications to the favorite sites of the walk and to the maximum of the local time. 1
Diffusion in random environment and the renewal theorem
 Ann. Probab
, 2003
"... Abstract. According to a theorem of S. Schumacher and T. Brox, for a diffusion X in a Brownian environment it holds that (Xt − blog t)/log 2 t → 0 in probability, as t → ∞, where b · is a stochastic process having an explicit description and depending only on the environment. In the first part of th ..."
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Cited by 7 (1 self)
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Abstract. According to a theorem of S. Schumacher and T. Brox, for a diffusion X in a Brownian environment it holds that (Xt − blog t)/log 2 t → 0 in probability, as t → ∞, where b · is a stochastic process having an explicit description and depending only on the environment. In the first part of this paper we compute the distribution of the sign changes for b on an interval [1, x] and study some of the consequences of the computation; in particular we get the probability of b keeping the same sign on that interval. These results have been announced in 1999 in a nonrigorous paper by P. Le Doussal, C. Monthus, and D. Fisher and were treated with a Renormalization Group analysis. We prove that this analysis can be made rigorous using a path decomposition for the Brownian environment and renewal theory. In the second part we consider the case that the environment is a spectrally one sided stable process and derive results describing the features of the environment that matter for the study of the process b. In particular we derive the distribution of b1. 1.
Diffusion at the random matrix hard edge
 Communications in Mathematical Physics 288 (2009
"... We show that the limiting minimal eigenvalue distributions for a natural generalization of Gaussian samplecovariance structures (the “beta ensembles”) are described in by the spectrum of a random diffusion generator. By a Riccati transformation, we obtain a second diffusion description of the limit ..."
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Cited by 6 (3 self)
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We show that the limiting minimal eigenvalue distributions for a natural generalization of Gaussian samplecovariance structures (the “beta ensembles”) are described in by the spectrum of a random diffusion generator. By a Riccati transformation, we obtain a second diffusion description of the limiting eigenvalues in terms of hitting laws. This picture pertains to the socalled hard edge of random matrix theory and sits in complement to the recent work [15] of the authors and B. Virág on the general beta random matrix soft edge. In fact, the diffusion descriptions found on both sides are used here to prove there exists a transition between the soft and hard edge laws at all values of beta. 1