Potential Analysis

, Volume 46, Issue 1, pp 167–180 | Cite as

Darning and Gluing of Diffusions

  • Wolfhard Hansen


We study darning of compact sets (darning and gluing of finite unions of compact sets), which are not thin at any of their points, in a potential-theoretic framework which may be described, analytically, in terms of harmonic kernels/harmonic functions or, probabilistically, in terms of a diffusion. This is accomplished without leaving our kind of setting so that the procedure can be iterated without any problem. It applies to darning and gluing of compacts in Euclidean spaces (manifolds) of different dimensions, which is of interest pertaining to recent studies on heat kernels.


Diffusion Brownian motion Harmonic function Harmonic kernel Darning Gluing Stable compact 

Mathematics Subject Classification (2010)

60J60 60J45 60J65 31A05 31D05 


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  1. 1.
    Bauer, H.: Šilovscher Rand und Dirichletsches Problem. Ann. Inst. Fourier Grenoble 11, 89–136, XIV (1961)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Bauer, H.: Harmonische Räume Und ihre Potentialtheorie. Lecture Notes in Mathematics 22. Springer, Berlin (1966)CrossRefGoogle Scholar
  3. 3.
    Bliedtner, J., Hansen, W.: Potential Theory – An Analytic and Probabilistic Approach to Balayage. Universitext. Springer, Berlin (1986)MATHGoogle Scholar
  4. 4.
    Chen, Z.-Q.: Topics on Recent Developments in the Theory of Markov Processes. Lectures at RIMS, Kyoto University (2012)Google Scholar
  5. 5.
    Chen, Z.-Q., Fukushima, M.: Symmetric Markov Processes, Time Change, and Boundary Theory. Princeton University Press (2012)Google Scholar
  6. 6.
    Chen, Z.-Q., Fukushima, M.: Stochastic Komatu-Loewner evolutions and BMD Domains constant. arXiv:1410.8257 (2013)
  7. 7.
    Chen, Z.-Q., Fukushima, M., Rhode, S.: Chordal Komatu-Loewner equations and Brownian motion with darning domains. Trans. Amer. Math. Soc. 368, 4065–4114 (2016)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Chen, Z.-Q., Lou, S.: Brownian motion on spaces with varying dimension. arXiv:1604.07870
  9. 9.
    Constantinescu, C., Cornea, A.: Potential Theory on Harmonic Spaces. Grundlehren D. Math. Wiss. Springer, Berlin (1972)CrossRefMATHGoogle Scholar
  10. 10.
    Grigor’yan, A., Hansen, W.: A Liouville property for Schrödinger operators. Math. Ann. 312, 659–716 (1998)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Hansen, W.: Perturbation of harmonic spaces and construction of semigroups. Invent. Math. 19, 149–164 (1973)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Hansen, W.: Some remarks on strict potentials. Math. Z. 147, 279–285 (1976)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Hansen, W.: Modification of balayage spaces by transitions with applications to coupling of PDE’s. Nagoya Math. J. 169, 77–118 (2003)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Hervé, R.-M.: Recherches axiomatiques sur la théorie des fonctions surharmoniques et du potentiel. Ann. Inst. Fourier 12, 415–517 (1962)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  1. 1.Fakultät für MathematikUniversität BielefeldBielefeldGermany

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