Displaying similar documents to “Central limit theorems for the brownian motion on large unitary groups”

The brownian cactus I. Scaling limits of discrete cactuses

Nicolas Curien, Jean-François Le Gall, Grégory Miermont (2013)

Annales de l'I.H.P. Probabilités et statistiques

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The cactus of a pointed graph is a discrete tree associated with this graph. Similarly, with every pointed geodesic metric space E , one can associate an -tree called the continuous cactus of E . We prove under general assumptions that the cactus of random planar maps distributed according to Boltzmann weights and conditioned to have a fixed large number of vertices converges in distribution to a limiting space called the Brownian cactus, in the Gromov–Hausdorff sense. Moreover, the Brownian...

Superdiffusivity for brownian motion in a poissonian potential with long range correlation II: Upper bound on the volume exponent

Hubert Lacoin (2012)

Annales de l'I.H.P. Probabilités et statistiques

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This paper continues a study on trajectories of Brownian Motion in a field of soft trap whose radius distribution is unbounded. We show here that for both point-to-point and point-to-plane model the volume exponent (the exponent associated to transversal fluctuation of the trajectories) ξ is strictly less than 1 and give an explicit upper bound that depends on the parameters of the problem. In some specific cases, this upper bound matches the lower bound proved in the first part of this...

The Dyson Brownian Minor Process

Mark Adler, Eric Nordenstam, Pierre Van Moerbeke (2014)

Annales de l’institut Fourier

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Consider an n × n Hermitean matrix valued stochastic process { H t } t 0 where the elements evolve according to Ornstein-Uhlenbeck processes. It is well known that the eigenvalues perform a so called Dyson Brownian motion, that is they behave as Ornstein-Uhlenbeck processes conditioned never to intersect. In this paper we study not only the eigenvalues of the full matrix, but also the eigenvalues of all the principal minors. That is, the eigenvalues of the k × k minors in the upper left corner...

Optimal stopping with advanced information flow: selected examples

Yaozhong Hu, Bernt Øksendal (2008)

Banach Center Publications

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We study optimal stopping problems for some functionals of Brownian motion in the case when the decision whether or not to stop before (or at) time t is allowed to be based on the δ-advanced information t + δ , where s is the σ-algebra generated by Brownian motion up to time s, s ≥ -δ, δ > 0 being a fixed constant. Our approach involves the forward integral and the Malliavin calculus for Brownian motion.

Superdiffusivity for brownian motion in a poissonian potential with long range correlation I: Lower bound on the volume exponent

Hubert Lacoin (2012)

Annales de l'I.H.P. Probabilités et statistiques

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We study trajectories of d -dimensional Brownian Motion in Poissonian potential up to the hitting time of a distant hyper-plane. Our Poissonian potential V is constructed from a field of traps whose centers location is given by a Poisson Point Process and whose radii are IID distributed with a common distribution that has unbounded support; it has the particularity of having long-range correlation. We focus on the case where the law of the trap radii ν has power-law decay and prove that...

Minkowski sums and Brownian exit times

Christer Borell (2007)

Annales de la faculté des sciences de Toulouse Mathématiques

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If C is a domain in n , the Brownian exit time of C is denoted by T C . Given domains C and D in n this paper gives an upper bound of the distribution function of T C + D when the distribution functions of T C and T D are known. The bound is sharp if C and D are parallel affine half-spaces. The paper also exhibits an extension of the Ehrhard inequality

Images of Gaussian random fields: Salem sets and interior points

Narn-Rueih Shieh, Yimin Xiao (2006)

Studia Mathematica

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Let X = X ( t ) , t N be a Gaussian random field in d with stationary increments. For any Borel set E N , we provide sufficient conditions for the image X(E) to be a Salem set or to have interior points by studying the asymptotic properties of the Fourier transform of the occupation measure of X and the continuity of the local times of X on E, respectively. Our results extend and improve the previous theorems of Pitt [24] and Kahane [12,13] for fractional Brownian motion.

Generators of Brownian motions on abstract Wiener spaces

Kei Harada (2010)

Banach Center Publications

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We prove that Brownian motion on an abstract Wiener space B generates a locally equicontinuous semigroup on C b ( B ) equipped with the T t -topology introduced by L. Le Cam. Hence we obtain a “Laplace operator” as its infinitesimal generator. Using this Laplacian, we discuss Poisson’s equation and heat equation, and study its properties, especially the difference from the Gross Laplacian.

Finite time asymptotics of fluid and ruin models: multiplexed fractional Brownian motions case

Krzysztof Dębicki, Grzegorz Sikora (2011)

Applicationes Mathematicae

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Motivated by applications in queueing fluid models and ruin theory, we analyze the asymptotics of ( s u p t [ 0 , T ] ( i = 1 n λ i B H i ( t ) - c t ) > u ) , where B H i ( t ) : t 0 , i = 1,...,n, are independent fractional Brownian motions with Hurst parameters H i ( 0 , 1 ] and λ₁,...,λₙ > 0. The asymptotics takes one of three different qualitative forms, depending on the value of m i n i = 1 , . . . , n H i .

An application of fine potential theory to prove a Phragmen Lindelöf theorem

Terry J. Lyons (1984)

Annales de l'institut Fourier

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We give a new proof of a Phragmén Lindelöf theorem due to W.H.J. Fuchs and valid for an arbitrary open subset U of the complex plane: if f is analytic on U , bounded near the boundary of U , and the growth of j is at most polynomial then either f is bounded or U { | z | > r } for some positive r and f has a simple pole.

Existence and asymptotic behaviour of some time-inhomogeneous diffusions

Mihai Gradinaru, Yoann Offret (2013)

Annales de l'I.H.P. Probabilités et statistiques

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Let us consider a solution of a one-dimensional stochastic differential equation driven by a standard Brownian motion with time-inhomogeneous drift coefficient ρ sgn ( x ) | x | α / t β . This process can be viewed as a Brownian motion evolving in a potential, possibly singular, depending on time. We prove results on the existence and uniqueness of solution, study its asymptotic behaviour and made a precise description, in terms of parameters ρ , α and β , of the recurrence, transience and convergence. More precisely,...

Potential theory of hyperbolic Brownian motion in tube domains

Grzegorz Serafin (2014)

Colloquium Mathematicae

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Let X = X(t); t ≥ 0 be the hyperbolic Brownian motion on the real hyperbolic space ℍⁿ = x ∈ ℝⁿ:xₙ > 0. We study the Green function and the Poisson kernel of tube domains of the form D × (0,∞)⊂ ℍⁿ, where D is any Lipschitz domain in n - 1 . We show how to obtain formulas for these functions using analogous objects for the standard Brownian motion in 2 n . We give formulas and uniform estimates for the set D a = x : x ( 0 , a ) . The constants in the estimates depend only on the dimension of the space. ...

Hitting distributions of geometric Brownian motion

T. Byczkowski, M. Ryznar (2006)

Studia Mathematica

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Let τ be the first hitting time of the point 1 by the geometric Brownian motion X(t) = x exp(B(t) - 2μt) with drift μ ≥ 0 starting from x > 1. Here B(t) is the Brownian motion starting from 0 with EB²(t) = 2t. We provide an integral formula for the density function of the stopped exponential functional A ( τ ) = 0 τ X ² ( t ) d t and determine its asymptotic behaviour at infinity. Although we basically rely on methods developed in [BGS], the present paper covers the case of arbitrary drifts μ ≥ 0 and provides...

The number of absorbed individuals in branching brownian motion with a barrier

Pascal Maillard (2013)

Annales de l'I.H.P. Probabilités et statistiques

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We study supercritical branching Brownian motion on the real line starting at the origin and with constant drift c . At the point x g t ; 0 , we add an absorbing barrier, i.e. individuals touching the barrier are instantly killed without producing offspring. It is known that there is a critical drift c 0 , such that this process becomes extinct almost surely if and only if c c 0 . In this case, if Z x denotes the number of individuals absorbed at the barrier, we give an asymptotic for P ( Z x = n ) as n goes to infinity....