Contact (Karlsruhe)
Prof. Dr. Günter Last
Karlsruhe Institute of Technology
Institut für Stochastik
Kaiserstraße 89
76133 Karlsruhe
Germany
Phone: +49-721-608 43265
Fax: +49-721-608 46691
Contact (Erlangen)
Prof. Dr. Klaus Mecke
Universität Erlangen-Nürnberg
Institut für Theoretische Physik
Staudtstraße 7
91058 Erlangen
Germany
Phone: +49-9131-85 28442
Fax: +49-9131-85 28444
Contact (Aarhus)
CSGB
Department of Mathematical Sciences
Ny Munkegade 118
building 1530
8000 Aarhus C
Denmark

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3. Boolean models

Boolean models are a fundamental object of stochastic geometry and continuum percolation and have applications in physics, materials science and biology, for example. A Boolean model is a random closed set which is the union of compact particles of a particle process. Often it is assumed that the particle process is stationary and generated by a Poisson process, but it is of significant interest to weaken these assumptions. The goal of this project is to gain deeper insight into the geometric properties of Boolean models and the dependence on the underlying particle process by considering various functionals of Boolean models. Examples are the intrinsic volumes and Minkowski tensors as well as non-additive functionals with local extensions and flag measures. Besides mean value and density formulas this project deals with the systematical treatment of second order properties and central limit theorems. A further topic are statistical methods to estimate characteristic quantities of Boolean models such as the distribution of radii of a spherical Boolean model.

Project Members

Cooperating partners

Publications

2018

  • Hug, Daniel and Rataj, Jan and Weil, Wolfgang
  • Flag representations of mixed volumes and mixed functionals of convex bodies
  • J. Math. Anal. Appl. 460(2) (2018)
  • [BibTeX]
  • Gieringer, Fabian and Last, Günter
  • Concentration inequalities for measures of a Boolean model
  • ALEA, Lat. Am. J. Probab. Math. Stat. 15, 151-166 (2018)
  • [Link] · [BibTeX]
  • Last, Günter and Nestmann, Franz and Schulte, Matthias
  • The random connection model and functions of edge-marked Poisson processes: second order properties and normal approximation
  • Preprint (2018)
    note: arXiv: 1808.01203
  • [arXiv] · [BibTeX]
  • Lachiéze-Rey, Raphaël and Schulte, Matthias and Yukich, J. E.
  • Normal approximation for stabilizing functionals
  • Annals of Applied Probability (to appear) (2018)
    note: arXiv: 1702.00726
  • [Link] · [BibTeX]

2017

  • Goodey, Paul and Hinderer, Wolfram and Hug, Daniel and Rataj, Jan and Weil, Wolfgang
  • A flag representation of projection functions
  • Adv. Geom. 17(3), 303–322 (2017)
  • [BibTeX]
  • Hörrmann, Julia and Svane, Anne Marie
  • Local Digital Algorithms Applied to Boolean Models
  • Scand. J. Stat. 44(2), 369–395 (2017)
  • [Link] · [BibTeX]
  • Schulte, Julia and Weil, Wolfgang
  • Valuations and Boolean models
  • pages 301–338 in: Lecture Notes in Math., Vol. 2177: Tensor valuations and their applications in stochastic geometry and imaging (editor(s): Jensen, Eva B. Vedel and Kiderlen, Markus), Springer, Cham, 2017
  • [BibTeX]
  • Hug, Daniel and Klatt, Michael A. and Last, Günter and Schulte, Matthias
  • Second Order Analysis of Geometric Functionals of Boolean Models
  • pages 339–383 in: Lecture Notes in Mathematics, Vol. 2177: Tensor Valuations and Their Applications in Stochastic Geometry and Imaging (editor(s): Vedel Jensen, Eva B. and Kiderlen, Markus), Springer International Publishing, Cham 2017
  • [Link] · [BibTeX]
  • Klatt, Michael A. and Schröder-Turk, Gerd E. and Mecke, Klaus
  • Mean-intercept anisotropy analysis of porous media. I. Analytic formulae for anisotropic Boolean models
  • Med. Phys. 44(7), 3650–3662 (2017)
  • [Link] · [BibTeX]
  • Klatt, Michael A. and Schröder-Turk, Gerd E. and Mecke, Klaus
  • Mean-intercept anisotropy analysis of porous media. II. Conceptual shortcomings of the MIL tensor definition and Minkowski tensors as an alternative
  • Med. Phys. 44(7), 3663–3675 (2017)
  • [Link] · [BibTeX]
  • Last, Günter and Penrose, Mathew D. and Zuyev, Sergei
  • On the capacity functional of the infinite cluster of a Boolean model
  • Ann. Appl. Probab. 27, 1678–1701 (2017)
  • [Link] · [BibTeX]
  • Last, Günter and Ziesche, Sebastian
  • On the Ornstein–Zernike equation for stationary cluster processes and the random connection model
  • Adv. Appl. Probab. 49(4), 1260–1287 (2017)
  • [BibTeX]
  • Reitzner, Matthias and Schulte, Matthias and Thäle, Christoph
  • Limit theory for the Gilbert graph
  • Adv. in Appl. Math. 88, 26–61 (2017)
  • [Link] · [arXiv] · [BibTeX]
  • Wolfgang Weil
  • Integral geometry of translation invariant functionals II: The case of general convex bodies
  • Adv. in Appl. Math. 83, 145–171 (2017)
  • [BibTeX]
  • Hug, Daniel and Weil, Wolfgang
  • Determination of Boolean models by mean values of mixed volumes
  • Preprint (2017)
    note: arXiv: 1712.08241
  • [arXiv] · [BibTeX]
  • Last, Günter and Penrose, Mathew
  • Lectures on the Poisson Process
  • Institute of Mathematical Statistics Textbooks: Cambridge University Press, Cambridge 2017
  • [Link] · [BibTeX]
  • Schulte, Matthias and Thäle, Christoph
  • Central limit theorems for the radial spanning tree
  • Random Structures & Algorithms 50(2), 262–286 (2017)
  • [Link] · [BibTeX]
  • Reitzner, Matthias and Schulte, Matthias and Thäle, Christoph
  • Limit theory for the Gilbert graph
  • Adv. Appl. Math. 88, 26–61 (2017)
  • [BibTeX]

2016

  • Hug, Daniel and Reitzner, Matthias
  • Introduction to stochastic geometry
  • pages 145–184 in: Bocconi Springer Ser., Vol. 7: Stochastic analysis for Poisson point processes (editor(s): Peccati, Giovanni and Reitzner, Matthias), Bocconi Univ. Press, [place of publication not identified], 2016
  • [BibTeX]
  • Daniel Hug, Günter Last and Matthias Schulte
  • Second-order properties and central limit theorems for geometric functionals of Boolean models
  • Ann. Appl. Probab. 26(1), 73–135 (2016)
  • [Link] · [BibTeX]
  • Klatt, Michael A.
  • Morphometry of random spatial structures in physics
  • FAU University Press, Erlangen 2016
    note: PhD thesis, Friedrich-Alexander-Universität Erlangen-Nürnberg
  • [BibTeX]
  • Ziesche, Sebastian
  • Perkolation auf zufälligen Mosaiken und im Boole'schen Modell
  • PhD thesis, Karlsruhe Institute of Technology (KIT), Karlsruhe 2016
  • [BibTeX]
  • Last, Günter
  • Stochastic Analysis for Poisson Processes
  • pages 1–36 in: Stochastic Analysis for Poisson Point Processes: Malliavin Calculus, Wiener-Itô Chaos Expansions and Stochastic Geometry (editor(s): Peccati, Giovanni and Reitzner, Matthias), Springer International Publishing, Cham 2016
  • [BibTeX]
  • Schulte, Matthias and Thäle, Christoph
  • Poisson Point Process Convergence and Extreme Values in Stochastic Geometry
  • pages 255–294 in: Stochastic Analysis for Poisson Point Processes: Malliavin Calculus, Wiener-Itô Chaos Expansions and Stochastic Geometry (editor(s): Peccati, Giovanni and Reitzner, Matthias), Springer International Publishing, Cham 2016
  • [BibTeX]
  • Schulte, Matthias
  • Normal Approximation of Poisson Functionals in Kolmogorov Distance
  • J. Theoret. Probab. 29(1), 96–117 (2016)
  • [Link] · [BibTeX]

2015

  • Günter Last and Hermann Thorisson
  • Construction and characterization of stationary and mass-stationary random measures on R^d
  • Stochastic Process. Appl. 125(12), 4473–4488 (2015)
  • [Link] · [BibTeX]
  • Scholz, Christian and Wirner, Frank and Klatt, Michael A. and Hirneise, Daniel and Schröder-Turk, Gerd E. and Mecke, Klaus and Bechinger, Clemens
  • Direct relations between morphology and transport in Boolean models
  • Phys. Rev. E 92(4), 043023 (2015)
  • [Link] · [BibTeX]
  • Wolfgang Weil
  • Integral geometry of translation invariant functionals I: The polytopal case
  • Adv. in Appl. Math. 66, 46–79 (2015)
  • [arXiv] · [BibTeX]

2014

  • Julia Hörrmann, Daniel Hug, Michael Klatt, and Klaus Mecke
  • Minkowski tensor density formulas for Boolean models
  • Adv. in Appl. Math. 55, 48–85 (2014)
  • [arXiv] · [BibTeX]
  • Daniel Hug, Günter Last, Zbynek Pawlas, and Wolfgang Weil
  • Statistics for Poisson models of overlapping spheres
  • Adv. in Appl. Probab. 46, 937–962 (2014)
  • [arXiv] · [BibTeX]
  • Julia Hörrmann
  • The method of densities for non-isotropic Boolean models
  • PhD thesis, Karlsruhe Institute of Technology (KIT), Karlsruhe 2014
  • [BibTeX]
  • Svane, Anne Marie
  • Local digital estimators of intrinsic volumes for Boolean models and in the design based setting
  • Adv. in Appl. Probab. 46, 35–58 (2014)
  • [Link] · [BibTeX]
  • Schulte, Matthias and Thäle, Christoph
  • Distances Between Poisson k-Flats
  • Methodol. Comput. Appl. Probab. 16(2), 311–329 (2014)
  • [Link] · [BibTeX]

2013

  • Daniel Hug, Jan Rataj, and Wolfgang Weil
  • A product integral representation of mixed volumes of two convex bodies
  • Adv. Geom. 13, 633–662 (2013)
  • [arXiv] · [BibTeX]
  • Daniel Hug, Ines Türk, and Wolfgang Weil
  • Flag measures for convex bodies
  • Fields Institute Communications (eds. Monika Ludwig, Vitali D. Milman, Vladimir Oestov, Nicole Tomczak-Jaegermann) 68, 145–187 (2013)
  • [Link] · [BibTeX]

2011

  • Günter Last and Ryszard Szekli
  • Comparisons and asymptotics for empty space hazard functions of germ-grain models
  • Adv. in Appl. Probab. 43, 942–962 (2011)
  • [BibTeX]