Integrable system of the heat kernel associated with logarithmic potentials

Kazuhiko Aomoto

Annales Polonici Mathematici (2000)

  • Volume: 74, Issue: 1, page 51-64
  • ISSN: 0066-2216

Abstract

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The heat kernel of a Sturm-Liouville operator with logarithmic potential can be described by using the Wiener integral associated with a real hyperplane arrangement. The heat kernel satisfies an infinite-dimensional analog of the Gauss-Manin connection (integrable system), generalizing a variational formula of Schläfli for the volume of a simplex in the space of constant curvature.

How to cite

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Aomoto, Kazuhiko. "Integrable system of the heat kernel associated with logarithmic potentials." Annales Polonici Mathematici 74.1 (2000): 51-64. <http://eudml.org/doc/208376>.

@article{Aomoto2000,
abstract = {The heat kernel of a Sturm-Liouville operator with logarithmic potential can be described by using the Wiener integral associated with a real hyperplane arrangement. The heat kernel satisfies an infinite-dimensional analog of the Gauss-Manin connection (integrable system), generalizing a variational formula of Schläfli for the volume of a simplex in the space of constant curvature.},
author = {Aomoto, Kazuhiko},
journal = {Annales Polonici Mathematici},
keywords = {Wiener integral; logarithmic potentials; Feynman-Kac formula; integrable system; heat kernel; logarithmic potential},
language = {eng},
number = {1},
pages = {51-64},
title = {Integrable system of the heat kernel associated with logarithmic potentials},
url = {http://eudml.org/doc/208376},
volume = {74},
year = {2000},
}

TY - JOUR
AU - Aomoto, Kazuhiko
TI - Integrable system of the heat kernel associated with logarithmic potentials
JO - Annales Polonici Mathematici
PY - 2000
VL - 74
IS - 1
SP - 51
EP - 64
AB - The heat kernel of a Sturm-Liouville operator with logarithmic potential can be described by using the Wiener integral associated with a real hyperplane arrangement. The heat kernel satisfies an infinite-dimensional analog of the Gauss-Manin connection (integrable system), generalizing a variational formula of Schläfli for the volume of a simplex in the space of constant curvature.
LA - eng
KW - Wiener integral; logarithmic potentials; Feynman-Kac formula; integrable system; heat kernel; logarithmic potential
UR - http://eudml.org/doc/208376
ER -

References

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  10. [10] M. Kac, Integration in Function Spaces and Some of Its Applications, Scuola Norm. Sup., Pisa, 1980. Zbl0504.28015
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  12. [12] M. L. Mehta, Random Matrices, 2nd ed., Academic Press, Boston, 1991. 
  13. [13] J. Milnor, The Schläfli differential equality, in: Collected Papers I: Geometry, Publish or Perish, Houston, TX, 1994, 281-295. 
  14. [14] P. Orlik and H. Terao, Arrangements of Hyperplanes, Springer, 1992. 
  15. [15] L. Schläfli, On the multiple integral ∫∫...∫ whose limits are p 1 = a 1 x + b 1 y + . . . + h 1 z ≥ 0 and x 2 + y 2 + . . . + z 2 = 1 , Quart. J. Math. 3 (1860), 54-68, 97-108. 
  16. [16] S. Watanabe, Analysis of Wiener functionals (Malliavin calculus) and its applications to heat kernels, Ann. Probab. 15 (1987), 1-39. Zbl0633.60077

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