Distributions of truncations of the heat kernel on the complex projective space

Demni, Nizar

Annales mathématiques Blaise Pascal, Tome 21 (2014) no. 2, pp. 1-20.

Let ${\left(U_t\right)}_{t\ge 0}$ be a Brownian motion valued in the complex projective space $ℂP^{N-1}$. Using unitary spherical harmonics of homogeneous degree zero, we derive the densities of $|U_t^1{|}^2$ and of $\left(\right|U_t^1{|}^2,\left|U_t^2{|}^2\right)$, and express them through Jacobi polynomials in the simplices of $ℝ$ and ${ℝ}^2$ respectively. More generally, the distribution of $\left(\right|U_t^1{|}^2,\cdots ,\left|U_t^k{|}^2\right),2\le k\le N-1$ 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 $1\le k\le N-2$, integrations by parts performed on the pde lead to a heat equation in the simplex of ${ℝ}^k$.

[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