Distributions of truncations of the heat kernel on the complex projective space
Annales mathématiques Blaise Pascal, Tome 21 (2014) no. 2, pp. 1-20.

Let (U t ) t0 be a Brownian motion valued in the complex projective space P N-1 . Using unitary spherical harmonics of homogeneous degree zero, we derive the densities of |U t 1 | 2 and of (|U t 1 | 2 ,|U t 2 | 2 ), and express them through Jacobi polynomials in the simplices of and 2 respectively. More generally, the distribution of (|U t 1 | 2 ,,|U t k | 2 ),2kN-1 may be derived using the decomposition of the unitary spherical harmonics under the action of the unitary group 𝒰(N-k+1) yet computations become tedious. We also revisit the approach initiated in [13] and based on a partial differential equation (hereafter pde) satisfied by the Laplace transform of the density. When k=1, we invert the Laplace transform and retrieve the expression already derived using spherical harmonics. For general 1kN-2, integrations by parts performed on the pde lead to a heat equation in the simplex of k .

DOI : 10.5802/ambp.339
Mots clés : Brownian motion, complex projective space, Dirichlet distribution, Jacobi polynomials in the simplex
Demni, Nizar 1

1 Institut de Recherche en Mathématiques de Rennes Université de Rennes 1 Campus de Beaulieu 35042 Rennes FRANCE
@article{AMBP_2014__21_2_1_0,
     author = {Demni, Nizar},
     title = {Distributions of truncations of the heat kernel on the complex projective space},
     journal = {Annales math\'ematiques Blaise Pascal},
     pages = {1--20},
     publisher = {Annales math\'ematiques Blaise Pascal},
     volume = {21},
     number = {2},
     year = {2014},
     doi = {10.5802/ambp.339},
     mrnumber = {3322612},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/ambp.339/}
}
TY  - JOUR
AU  - Demni, Nizar
TI  - Distributions of truncations of the heat kernel on the complex projective space
JO  - Annales mathématiques Blaise Pascal
PY  - 2014
SP  - 1
EP  - 20
VL  - 21
IS  - 2
PB  - Annales mathématiques Blaise Pascal
UR  - http://www.numdam.org/articles/10.5802/ambp.339/
DO  - 10.5802/ambp.339
LA  - en
ID  - AMBP_2014__21_2_1_0
ER  - 
%0 Journal Article
%A Demni, Nizar
%T Distributions of truncations of the heat kernel on the complex projective space
%J Annales mathématiques Blaise Pascal
%D 2014
%P 1-20
%V 21
%N 2
%I Annales mathématiques Blaise Pascal
%U http://www.numdam.org/articles/10.5802/ambp.339/
%R 10.5802/ambp.339
%G en
%F AMBP_2014__21_2_1_0
Demni, Nizar. Distributions of truncations of the heat kernel on the complex projective space. Annales mathématiques Blaise Pascal, Tome 21 (2014) no. 2, pp. 1-20. doi : 10.5802/ambp.339. http://www.numdam.org/articles/10.5802/ambp.339/

[1] Aktas, R.; Xu, Y. Sobolev orthogonal polynomials on a simplex, Int. Math. Res. Notice, Volume 13 (2013), pp. 3087-3131 | MR

[2] Andrews, G. E.; Askey, R.; Roy, R. Special functions, Cambridge University Press, Cambridge, 1999 | MR | Zbl

[3] Bakry, D. Remarques sur les semi-groupes de Jacobi, Hommage à P. André Meyer et J. Neveu. Astérisque, Volume 236 (1996), pp. 23-39 | Numdam | MR | Zbl

[4] Benabdallah, A. Noyau de diffusion sur les espaces homogènes compacts, Bull. Soc. Math. France, Volume 101 (1973), pp. 265-283 | Numdam | MR | Zbl

[5] Berger, M.; Gauduchon, P.; Mazet, E. Le spectre d’une variété Riemannienne, Lecture Notes in Mathematics, Springer-Verlag, 1971, pp. vii+251 pp | MR | Zbl

[6] Dunkl, C.; Xu, Y. Orthogonal Polynomials of Several Variables, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2001 | MR | Zbl

[7] Gasper, G. Banach algebras for Jacobi series and positivity of a kernel, Ann. Math., Volume 95 (1972), pp. 261-280 | DOI | MR | Zbl

[8] Grinberg, E. L. Spherical harmonics and integral geometry on projective spaces, Trans. Amer. Math. Soc., Volume 279 (1983), pp. 187-203 | DOI | MR | Zbl

[9] Hiai, F.; Petz, D. The Semicircle Law, Free Random Variables and Entropy, 77, A. M. S., 2000, pp. x+376 pp | MR | Zbl

[10] Karlin, S. P.; McGregor, G. Classical diffusion processes and total positivity, J. Math. Anal. Appl., Volume 1 (1960), pp. 163-183 | DOI | MR | Zbl

[11] Koornwinder, T. The addition formula for Jacobi polynomials II. The Laplace type integral and the product formula, Report TW 133/72. Mathematisch Centrum, Amsterdam (1972) | Zbl

[12] Koornwinder, T. The addition formula for Jacobi polynomials III. Completion of the proof, Report TW 135/72. Mathematisch Centrum, Amsterdam (1972) | Zbl

[13] Nechita, I.; Pellegrini, C. Random pure quantum states via unitary Brownian motion, Electron. Commun. Probab, Volume 18 (2013), pp. 1-13 | DOI | MR

[14] Vilenkin, N. Ya. Fonctions spéciales et théorie de la représentation des groupes, 33, Monographies Universitaires de Mathématiques, Dunod, 1969 | Zbl

[15] Watson, G. N. A treatise on the theory of Bessel functions, Cambridge University Press, 1995 | MR | Zbl

Cité par Sources :