Closed form formulae for the heat kernels and the Green functions for the Laplacians on the symmetric spaces of rank one

Bulletin des Sciences Mathematiques

Volumn 125, Issue 6-7, 2001, Pages 553-581

  Matsumoto, Hiroyuki ^a  

^a NAGOYA UNIVERSITY   (Japan)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0345878075     PISSN: 00074497     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0007-4497(01)01099-5     Document Type: Article

References (30)

1. Alili L., Matsumoto H., Shiraishi T. On a triplet of exponential Brownian functionals. Sém. Prob. XXXV. Lecture Notes in Math. 1755:2001;396-415 Springer, Berlin.
2. Borel A., Casselman W.C. (Eds.), Automorphic Forms, Representations, and L -functions, Proc. Symp. Pure Math., Vol. 33, Amer. Math. Soc., Providence, RI, 1979.
3. Davies E.B. Heat Kernels and Spectral Theory. 1989;Cambridge Univ. Press, Cambridge.
4. e série. 103:1979;305-351.
5. Debiard A., Gaveau B. Analysis on root systems. Can. J. Math. 39:1987;1281-1404.
6. Elstrodt J. Die Resolvente zum Eigenwertproblem der automorphen Formen in der hyperbolischen Ebene, Teil I. Math. Ann. 203:1973;295-330. Teil II, Math. Z., 132 (1973), 99-134; Teil III, Math. Ann. 208 (1974), 99-132.
7. Fay J. Fourier coefficients of the resolvent for a Fuchsian group. J. Reine Angew. Math. 293:1977;143-203.
8. Gangolli R. The length spectra of some compact manifolds of negative curvature. J. Differential Geom. 12:1977;403-424.
9. Gangolli R. Zeta functions of Selberg's type for compact space forms of symmetric spaces of rank one. Illinois J. Math. 21:1977;1-41.
10. Gradshteyn I.S., Ryzhik I.M., Tables of Integrals, Series and Products, 6th edn., Jeffrey, A. (Ed.), Academic Press, New York, 2000.
11. Grosche C., Steiner F. Handbook of Feynman Path Integrals. 1998;Springer, Berlin.
12. Gruet J.-C. Semi-groupe du mouvement Brownien hyperbolique. Stochastics Stochastic Rep. 56:1996;53-61.
13. Gruet J.-C., Windings of hyperbolic Brownian motion, in [30].
14. Hartman P., Watson G.S. "Normal" distribution functions on spheres and the modified Bessel functions. Ann. Probab. 2:1974;593-607.
15. Hejhal D. The Selberg Trace Formula for PSL(2,R) , Vol. 1. Lecture Notes in Math. 548:1976;Springer, Berlin.
16. Hejhal D. The Selberg Trace Formula for PSL(2,R) , Vol. 2. Lecture Notes in Math. 1001:1983;Springer, Berlin.
17. Ikeda N., Matsumoto H. Brownian motion on the hyperbolic plane and Selberg trace formula. J. Func. Anal. 163:1999;63-110.
18. Ikeda N., Watanabe S. Stochastic Differential Equations and Diffusion Processes. 2nd edn. 1989;North-Holland/Kodansha, Amsterdam/Tokyo.
19. Lohoué N., Rychener T. Die Resolvente von Δ auf symmetrischen Räumen vom nichtkompakten Typ. Comment. Math. Helvetici. 57:1982;445-468.
20. Malliavin P. Stochastic calculus of variation and hypoelliptic operators. Itô K. Proc. Internat. Symp. SDE Kyoto, 1976. 1978;195-263 Kinokuniya, Tokyo.
21. Matsumoto H., Nguyen L., Yor M., Subordinators related to the exponential functionals of Brownian bridges and explicit formulae for the semigroups of hyperbolic Brownian motions, in: Engelbert H., Buckdahn R., Yor M. (Eds.) Proceedings of the 2000 Winter School in Siegmundsburg, Gordon and Breach Publ., to appear.
22. Matsumoto H., Ueki N. Applications of the theory of the metaplectic representation to quadratic Hamiltonians on the two-dimensional Euclidean space. J. Math. Soc. Japan. 52:2000;269-292.
23. Moore C.C. (Ed.), Harmonic Analysis in Homogeneous Spaces, Proc. Symp. Pure Math., Vol. 26, Amer. Math. Soc., Providence, RI, 1973.
24. Selberg A. Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series. J. Indian Math. Soc. 20:1956;47-87.
25. Terras A. Harmonic Analysis on Symmetric Spaces and Applications I. 1985;Springer, Berlin.
26. Venkov A.B. Expansion in automorphic eigenfunctions of the Laplace-Beltrami operator in classical symmetric spaces of rank one, and the Selberg trace formula. Proc. Steklov Inst. Math. 125:1973;1-48.
27. Watanabe S. Analysis of Wiener functionals (Malliavin calculus) and its applications to heat kernels. Ann. Probab. 15:1987;1-39.
28. Yor M. Loi de l'indice du lacet Brownien, et distribution de Hartman-Watson. Zeits. Wahrscheinlichkeitsth. 53:1980;71-95.
29. Yor M. On some exponential functionals of Brownian motion. Adv. Appl. Probab. 24:1992;509-531.
30. Yor M. Exponential Functionals and Principal Values related to Brownian Motion, A Collection of Research Papers, Biblioteca de la Revista Matemática Iberoamericana, Madrid. 1997.