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Trees with random conductivities and the (reciprocal) inverse Gaussian distribution

Part of: Probability theory on algebraic and topological structures Special processes

Published online by Cambridge University Press:  01 July 2016

O. E. Barndorff-Nielsen  and
A. E. Koudou
O. E. Barndorff-Nielsen*
Affiliation:
Aarhus University
A. E. Koudou*
Affiliation:
Université Paul Sabatier
*
∗ Postal address: Dept of Mathematical Sciences, Aarhus University, DK-8000 Aarhus C, Denmark. Email address: [email protected]
∗∗ Postal address: Laboratoire de Statistique et Probabilités, Université Paul Sabatier, F-31062 Toulouse cedex, France.
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Abstract

Equipping the edges of a finite rooted tree with independent resistances that are inverse Gaussian for interior edges and reciprocal inverse Gaussian for terminal edges makes it possible, for suitable constellations of the parameters, to show that the total resistance is reciprocal inverse Gaussian (Barndorff-Nielsen 1994). This result is extended to infinite trees. Also, a connection to Brownian diffusion is established and, for the case of finite trees, an exact distributional and independence result is derived for the conditional model given the total resistance.

Keywords

Brownian diffusiongamma distributionhitting timesKirchhoff-Ohm laws

MSC classification

Primary: 60B99: None of the above, but in this section
Secondary: 60K99: None of the above, but in this section
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 30 , Issue 2 , June 1998 , pp. 409 - 424
Copyright
Copyright © Applied Probability Trust 1998 

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References

Arkhangel'skii, A. V. and Pontryagin, L. S. (1990). General Topology I. In Encyclopaedia of Mathematical Sciences 17. Springer, New York.Google Scholar
Barndorff-Nielsen, O. E. (1994). A note on electrical networks and the inverse Gaussian distribution. Adv. Appl. Prob. 26, 63–67.CrossRefGoogle Scholar
Bassingthwaighte, J. B., King, R. B. and Roger, S. A. (1989). Fractal nature of regional myocardial blood flow heterogeneity. Circulation Res. 65, 578–590.CrossRefGoogle ScholarPubMed
Bhattacharya, R. N. and Waymire, E. C. (1990). Stochastic Processes with Applications. Wiley, New York.Google Scholar
Grimmet, G. (1993). Random graphical networks. In Chaos and Networks. Statistical and Probabilistic Aspects. eds. Barndorff-Nielsen, O. E., Jensen, J. L. and Kendall, W. S.. Chapman and Hall, London, pp. 288–301.CrossRefGoogle Scholar
Kesten, H. (1982). Percolation Theory for Mathematicians. Birkhäuser, Basel.CrossRefGoogle Scholar
Letac, G. and Seshadri, V. (1995). A random continued fraction in Rd+1 with an inverse Gaussian distribution. Bernoulli 1, 381–393.CrossRefGoogle Scholar
Mayrovitz, H. N. and Roy, J. (1983). Micro-vascular blood flow: evidence indicating a cubic dependence on arteriolar diameter. Amer. J. Physiol. 245 (Heart Circ. Physiol. 14), H1031–H1038.Google Scholar
Mulvany, M. J. and Aalkjær, C. (1990). Structure and function of small arteries. Physiol. Rev. 70, 921–961.CrossRefGoogle ScholarPubMed
Pemantle, R. (1995). Tree-indexed processes. Statist. Sci. 10, 200–213.CrossRefGoogle Scholar
Sandau, K. and Kurz, H. (1994). Modelling of vascular growth processes: a stochastic biophysical approach to embryonic angiogenesis. J. Microscopy 175, 205–213.CrossRefGoogle ScholarPubMed
Schmid-Schönbein, G. W., Firestone, G. and Zweifach, B. W. (1986). Network anatomy of arteries feeding the spinotrapezius muscle in normotensive and hypertensive rats. Blood Vessels 23, 34–49.Google ScholarPubMed
Soardi, P. M. (1994). Potential Theory on Infinite Networks. Springer, Berlin.CrossRefGoogle Scholar
Sun, T., Meakin, P. and Jossang, T. (1995). Minimum energy dissipation and fractal structures of vascular systems. Fractals 3, 123–153.CrossRefGoogle Scholar
Vallois, P. (1991). La loi Gaussienne inverse généralisée comme premier ou dernier temps de passage de diffusion. Bull. Sci. Math. 2e Série 115, 301–368.Google Scholar
VanBavel, E. and Spaan, J. A. E. (1992). Branching patterns in the porcine coronary arterial tree. Estimation of flow heterogeneity. Circulation Res. 71, 1200–1212.CrossRefGoogle ScholarPubMed