Minkowski sums and Brownian exit times

Christer Borell

Displaying similar documents to “Minkowski sums and Brownian exit times”

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 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...

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 $m i n_{i = 1, . . ., n} H_{i}$ .

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 $U \supset {| z | > r}$ for some positive $r$ and $f$ has a simple pole.

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 $c \geq c_{0}$ . 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....

Images of Gaussian random fields: Salem sets and interior points

Narn-Rueih Shieh, Yimin Xiao (2006)

Studia Mathematica

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Let $X = X (t), t \in ℝ^{N}$ be a Gaussian random field in $ℝ^{d}$ with stationary increments. For any Borel set $E \subset ℝ^{N}$ , 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.

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...

Upper bounds for the density of solutions to stochastic differential equations driven by fractional brownian motions

Fabrice Baudoin, Cheng Ouyang, Samy Tindel (2014)

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

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In this paper we study upper bounds for the density of solution to stochastic differential equations driven by a fractional Brownian motion with Hurst parameter $H g t; 1 / 3$ . We show that under some geometric conditions, in the regular case $H g t; 1 / 2$ , the density of the solution satisfies the log-Sobolev inequality, the Gaussian concentration inequality and admits an upper Gaussian bound. In the rough case $H g t; 1 / 3$ and under the same geometric conditions, we show that the density of the solution is smooth and...

The generalized Holditch theorem for the homothetic motions on the planar kinematics

Nuri Kuruoğlu, Salim Yüce (2004)

Czechoslovak Mathematical Journal

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W. Blaschke and H. R. Müller [4, p. 142] have given the following theorem as a generalization of the classic Holditch Theorem: Let $E / E^{'}$ be a 1-parameter closed planar Euclidean motion with the rotation number $ν$ and the period $T$ . Under the motion $E / E^{'}$ , let two points $A = (0, 0)$ , $B = (a + b, 0) \in E$ trace the curves $k_{A}, k_{B} \subset E^{'}$ and let $F_{A}, F_{B}$ be their orbit areas, respectively. If $F_{X}$ is the orbit area of the orbit curve $k$ of the point $X = (a, 0)$ which is collinear with points $A$ and $B$ then $F_{X} = \frac{[a F_{B} + b F_{A}]}{a + b} - π ν a b .$ In this paper, under the 1-parameter closed planar homothetic...