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

Hubert Lacoin

Displaying similar documents to “Superdiffusivity for brownian motion in a poissonian potential with long range correlation I: Lower bound on the volume exponent”

Local percolative properties of the vacant set of random interlacements with small intensity

Alexander Drewitz, Balázs Ráth, Artëm Sapozhnikov (2014)

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

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Random interlacements at level $u$ is a one parameter family of connected random subsets of $ℤ^{d}$ , $d \geq 3$ ( (2010) 2039–2087). Its complement, the vacant set at level $u$ , exhibits a non-trivial percolation phase transition in $u$ ( (2009) 831–858; (2010) 2039–2087), and the infinite connected component, when it exists, is almost surely unique ( (2009) 454–466). In this paper we study local percolative properties of the vacant set of random...

Convergence rates for the full gaussian rough paths

Peter Friz, Sebastian Riedel (2014)

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

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Under the key assumption of finite $ρ$ -variation, $ρ \in [1, 2)$ , of the covariance of the underlying Gaussian process, sharp a.s. convergence rates for approximations of Gaussian rough paths are established. When applied to Brownian resp. fractional Brownian motion (fBM), $ρ = 1$ resp. $ρ = 1 / (2 H)$ , we recover and extend the respective results of ( (2009) 2689–2718) and ( (2012) 518–550). In particular, we establish an a.s. rate $k^{- (1 / ρ - 1 / 2 - ε)}$ , any $ε g t; 0$ , for Wong–Zakai and Milstein-type approximations...

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...

Ergodic behaviour of “signed voter models”

G. Maillard, T. S. Mountford (2013)

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

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We answer some questions raised by Gantert, Löwe and Steif ( (2005) 767–780) concerning “signed” voter models on locally finite graphs. These are voter model like processes with the difference that the edges are considered to be either positive or negative. If an edge between a site $x$ and a site $y$ is negative (respectively positive) the site $y$ will contribute towards the flip rate of $x$ if and only if the two current spin values are equal (respectively opposed). ...

Localization and delocalization for heavy tailed band matrices

Florent Benaych-Georges, Sandrine Péché (2014)

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

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We consider some random band matrices with band-width $N^{μ}$ whose entries are independent random variables with distribution tail in $x^{- α}$ . We consider the largest eigenvalues and the associated eigenvectors and prove the following phase transition. On the one hand, when $α l t; 2 (1 + μ^{- 1})$ , the largest eigenvalues have order $N^{(1 + μ) / α}$ , are asymptotically distributed as a Poisson process and their associated eigenvectors are essentially carried by two coordinates (this phenomenon has already been remarked for full matrices...

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 $ℝ$ -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...

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

Supercritical super-brownian motion with a general branching mechanism and travelling waves

A. E. Kyprianou, R.-L. Liu, A. Murillo-Salas, Y.-X. Ren (2012)

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

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We offer a probabilistic treatment of the classical problem of existence, uniqueness and asymptotics of monotone solutions to the travelling wave equation associated to the parabolic semi-group equation of a super-Brownian motion with a general branching mechanism. Whilst we are strongly guided by the reasoning in Kyprianou ( (2004) 53–72) for branching Brownian motion, the current paper offers a number of new insights. Our analysis incorporates the role of Seneta–Heyde...

Central limit theorems for the brownian motion on large unitary groups

Florent Benaych-Georges (2011)

Bulletin de la Société Mathématique de France

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In this paper, we are concerned with the large $n$ limit of the distributions of linear combinations of the entries of a Brownian motion on the group of $n \times n$ unitary matrices. We prove that the process of such a linear combination converges to a Gaussian one. Various scales of time and various initial distributions are considered, giving rise to various limit processes, related to the geometric construction of the unitary Brownian motion. As an application, we propose a very short proof of...

Characterizations of processes with stationary and independent increments under $G$ -expectation

Yongsheng Song (2013)

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

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Our purpose is to investigate properties for processes with stationary and independent increments under $G$ -expectation. As applications, we prove the martingale characterization of $G$ -Brownian motion and present a pathwise decomposition theorem for generalized $G$ -Brownian motion.

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 $ℱ_{t + δ}$ , where $ℱ_{s}$ 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.

The Dyson Brownian Minor Process

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

Annales de l’institut Fourier

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Consider an $n \times n$ Hermitean matrix valued stochastic process ${H_{t}}_{t \geq 0}$ 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 $k \times k$ minors in the upper left corner...

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 $ℙ (s u p_{t \in [0, T]} (\sum_{i = 1}^{n} λ_{i} B_{H_{i}} (t) - c t) > u)$ , where $B_{H_{i}} (t) : t \geq 0$ , i = 1,...,n, are independent fractional Brownian motions with Hurst parameters $H_{i} \in (0, 1]$ and λ₁,...,λₙ > 0. The asymptotics takes one of three different qualitative forms, depending on the value of $m i n_{i = 1, . . ., n} H_{i}$ .