Displaying similar documents to “Hitting half-spaces or spheres by Ornstein-Uhlenbeck type diffusions”

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.

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...

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.

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...