Duality and the Martin compactification

John C. Taylor

Annales de l'institut Fourier (1972)

  • Volume: 22, Issue: 3, page 95-130
  • ISSN: 0373-0956

Abstract

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Let ℋ be a Bauer sheaf that admits a Green function. Then there exists a diffusion process corresponding to the sheaf whose resolvent possesses a Hunt-Kunita-Watanabe dual resolvent that comes from a diffusion process. If ℋ is a Brelot sheaf which possesses an adjoint sheaf ℋ * the dual process corresponds to ℋ * .The Martin compactification defined by a Brelot sheaf ℋ that admits a Green function coincides with a Kunita-Watanabe compactification defined by the dual resolvent.

How to cite

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Taylor, John C.. "Duality and the Martin compactification." Annales de l'institut Fourier 22.3 (1972): 95-130. <http://eudml.org/doc/74095>.

@article{Taylor1972,
abstract = {Let $\{\cal H\}$ be a Bauer sheaf that admits a Green function. Then there exists a diffusion process corresponding to the sheaf whose resolvent possesses a Hunt-Kunita-Watanabe dual resolvent that comes from a diffusion process. If $\{\cal H\}$ is a Brelot sheaf which possesses an adjoint sheaf $\{\cal H\}^*$ the dual process corresponds to $\{\cal H\}^*$.The Martin compactification defined by a Brelot sheaf $\{\cal H\}$ that admits a Green function coincides with a Kunita-Watanabe compactification defined by the dual resolvent.},
author = {Taylor, John C.},
journal = {Annales de l'institut Fourier},
language = {eng},
number = {3},
pages = {95-130},
publisher = {Association des Annales de l'Institut Fourier},
title = {Duality and the Martin compactification},
url = {http://eudml.org/doc/74095},
volume = {22},
year = {1972},
}

TY - JOUR
AU - Taylor, John C.
TI - Duality and the Martin compactification
JO - Annales de l'institut Fourier
PY - 1972
PB - Association des Annales de l'Institut Fourier
VL - 22
IS - 3
SP - 95
EP - 130
AB - Let ${\cal H}$ be a Bauer sheaf that admits a Green function. Then there exists a diffusion process corresponding to the sheaf whose resolvent possesses a Hunt-Kunita-Watanabe dual resolvent that comes from a diffusion process. If ${\cal H}$ is a Brelot sheaf which possesses an adjoint sheaf ${\cal H}^*$ the dual process corresponds to ${\cal H}^*$.The Martin compactification defined by a Brelot sheaf ${\cal H}$ that admits a Green function coincides with a Kunita-Watanabe compactification defined by the dual resolvent.
LA - eng
UR - http://eudml.org/doc/74095
ER -

References

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  1. [1] H. BAUER, Harmonische Raüme und ihre Potentialtheorie, Lectures 22, Springer-Verlag, Berlin 1966. Zbl0142.38402
  2. [2] R. CAIROLI, Produits de semi-groupes de transition et produits de processus, Publ. Inst. Stat. Univ. Paris 15 (1966). Zbl0161.14603MR35 #7397
  3. [3] G. CHOQUET, Le problème des moments, Séminaire de l'Initiation à l'Analyse (Choquet) Institut Henri Poincaré 1ère année 1962. 
  4. [4] C. CONSTANTINESCU, A topology on the cone of non-negative superharmonic functions, Rev. Roum. Math. Pures et Appl. X (1965), 1331-1348. Zbl0144.36603MR35 #1804
  5. [5] C. CONSTANTINESCU, Kernels and nuclei on harmonic spaces, Rev. Roum. Math. Pures et Appl. XIII (1968), 35-37. Zbl0159.40901MR38 #332
  6. [6] W. HANSEN, Konstruktion von Halbgruppen und Markoffschen Prozessen, Inventiones Math. 3 (1967), 179-214. Zbl0158.12803MR38 #771
  7. [7] R.M. HERVE, Recherches axiomatiques sur la théorie des fonctions surharmoniques et du potentiel, Ann. Inst. Fourier 12 (1962), 415-571. Zbl0101.08103MR25 #3186
  8. [8] H. KUNITA, and T. WATANABE, Markov processes and Martin Boundaries, Part I, Illinois J. Math. 9 (1965), 485-526. Zbl0147.16505MR31 #5240
  9. [9] P.A. MEYER, Probabilités et potentiel, Hermann et Cie, Paris 1966. Zbl0138.10402MR34 #5118
  10. [10] P.A. MEYER, Processus de Markov, Lecture Notes 26, Springer-Verlag, Berlin 1967. Zbl0189.51403MR36 #2219
  11. [11] P.A. MEYER, Représentation intégrale des fonctions excessives, résultats de Mokobodzki, Séminaire de Probabilités V (Université de Strasbourg), Springer-Verlag, Berlin 1971. 
  12. [12] M. SIEVEKING, Integral darstellung superharmonischer Funktionen mit anwendung auf parabolische differentialgleichungen, Seminar uber potential theorie, Springer-Verlag, Berlin 1968. 
  13. [13] M. SIEVEKING, On the existence of a dual in axiomatic potential theory (unpublished manuscript). 
  14. [14] J.C. TAYLOR, The Martin boundaries of equivalent sheaves, Ann. Inst. Fourier XX (1970), 433-456. Zbl0185.19801MR42 #2022
  15. [15] J.C. TAYLOR, Balayage de fonctions excessives, Séminaire de Théorie du Potentiel (Brelot, Choquet, Deny). Institut Henri Poincaré 14ème année (1970-1971). Zbl0317.31010
  16. [16] J.C. TAYLOR, Strict potentials and Hunt processes, Inventiones math. 16 (1972), 249-559. Zbl0225.31013MR45 #9389

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