Displaying similar documents to “Poisson kernel and Green function of balls for complex hyperbolic Brownian motion”

Feynman-Kac formula, λ-Poisson kernels and λ-Green functions of half-spaces and balls in hyperbolic spaces

Tomasz Byczkowski, Jacek Małecki, Tomasz Żak (2010)

Colloquium Mathematicae

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We apply the Feynman-Kac formula to compute the λ-Poisson kernels and λ-Green functions for half-spaces or balls in hyperbolic spaces. We present known results in a unified way and also provide new formulas for the λ-Poisson kernels and λ-Green functions of half-spaces in ℍⁿ and for balls in real and complex hyperbolic spaces.

Limiting behaviors of the Brownian motions on hyperbolic spaces

H. Matsumoto (2010)

Colloquium Mathematicae

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Using explicit representations of the Brownian motions on hyperbolic spaces, we show that their almost sure convergence and the central limit theorems for the radial components as time tends to infinity can be easily obtained. We also give a straightforward strategy to obtain explicit expressions for the limit distributions or Poisson kernels.

Hitting half-spaces or spheres by Ornstein-Uhlenbeck type diffusions

Tomasz Byczkowski, Jakub Chorowski, Piotr Graczyk, Jacek Małecki (2012)

Colloquium Mathematicae

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The purpose of the paper is to provide a general method for computing the hitting distributions of some regular subsets D for Ornstein-Uhlenbeck type operators of the form 1/2Δ + F·∇, with F bounded and orthogonal to the boundary of D. As an important application we obtain integral representations of the Poisson kernel for a half-space and balls for hyperbolic Brownian motion and for the classical Ornstein-Uhlenbeck process. The method developed in this paper is based on stochastic calculus...

Sharp estimates of the Green function of hyperbolic Brownian motion

Kamil Bogus, Tomasz Byczkowski, Jacek Małecki (2015)

Studia Mathematica

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The main objective of the work is to provide sharp two-sided estimates of the λ-Green function, λ ≥ 0, of the hyperbolic Brownian motion of a half-space. We rely on the recent results obtained by K. Bogus and J. Małecki (2015), regarding precise estimates of the Bessel heat kernel for half-lines. We also substantially use the results of H. Matsumoto and M. Yor (2005) on distributions of exponential functionals of Brownian motion.

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...

Brownian motion and parabolic Anderson model in a renormalized Poisson potential

Xia Chen, Alexey M. Kulik (2012)

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

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A method known as renormalization is proposed for constructing some more physically realistic random potentials in a Poisson cloud. The Brownian motion in the renormalized random potential and related parabolic Anderson models are modeled. With the renormalization, for example, the models consistent to Newton’s law of universal attraction can be rigorously constructed.

On the analogy between self-gravitating Brownian particles and bacterial populations

Pierre-Henri Chavanis, Magali Ribot, Carole Rosier, Clément Sire (2004)

Banach Center Publications

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We develop the analogy between self-gravitating Brownian particles and bacterial populations. In the high friction limit, the self-gravitating Brownian gas is described by the Smoluchowski-Poisson system. These equations can develop a self-similar collapse leading to a finite time singularity. Coincidentally, the Smoluchowski-Poisson system corresponds to a simplified version of the Keller-Segel model of bacterial populations. In this biological context, it describes the chemotactic...

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 ℝ n - 1 . We show how to obtain formulas for these functions using analogous objects for the standard Brownian motion in ℝ 2 n . We give formulas and uniform estimates for the set D a = x ∈ ℍ ⁿ : x ₁ ∈ ( 0 , a ) . The constants in the estimates depend only on the dimension of the space. ...

Computation of Biharmonic Poisson Kernel for the Upper Half Plane

Ali Abkar (2007)

Bollettino dell'Unione Matematica Italiana

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We first consider the biharmonic Poisson kernel for the unit disk, and study the boundary behavior of potentials associated to this kernel function. We shall then use some properties of the biharmonic Poisson kernel for the unit disk to compute the analogous biharmonic Poisson kernel for the upper half plane.

Extremal problems for conditioned brownian motion and the hyperbolic metric

Rodrigo Bañuelos, Tom Carroll (2000)

Annales de l'institut Fourier

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This paper investigates isoperimetric-type inequalities for conditioned Brownian motion and their generalizations in terms of the hyperbolic metric. In particular, a generalization of an inequality of P. Griffin, T. McConnell and G. Verchota, concerning extremals for the lifetime of conditioned Brownian motion in simply connected domains, is proved. The corresponding lower bound inequality is formulated in various equivalent forms and a special case of these is proved.

Motion planning for a class of boundary controlled linear hyperbolic PDE's involving finite distributed delays

Frank Woittennek, Joachim Rudolph (2010)

ESAIM: Control, Optimisation and Calculus of Variations

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Motion planning and boundary control for a class of linear PDEs with constant coefficients is presented. With the proposed method transitions from rest to rest can be achieved in a prescribed finite time. When parameterizing the system by a flat output, the system trajectories can be calculated from the flat output trajectory by evaluating definite convolution integrals. The compact kernels of the integrals can be calculated using infinite series. Explicit formulae are derived...