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 ${\left({U}_{t}\right)}_{t\ge 0}$ be a Brownian motion valued in the complex projective space $ℂ{P}^{N-1}$. Using unitary spherical harmonics of homogeneous degree zero, we derive the densities of $|{U}_{t}^{1}{|}^{2}$ and of $\left(|{U}_{t}^{1}{|}^{2},|{U}_{t}^{2}{|}^{2}\right)$, and express them through Jacobi polynomials in the simplices of $ℝ$ and ${ℝ}^{2}$ respectively. More generally, the distribution of $\left(|{U}_{t}^{1}{|}^{2},\cdots ,|{U}_{t}^{k}{|}^{2}\right),\phantom{\rule{0.277778em}{0ex}}2\le k\le N-1$ may be derived using the decomposition of the unitary spherical harmonics under the action of the unitary group $𝒰\left(N-k+1\right)$ 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 ${ℝ}^{k}$.

DOI : https://doi.org/10.5802/ambp.339
Mots clés : Brownian motion, complex projective space, Dirichlet distribution, Jacobi polynomials in the simplex
@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 = {www.numdam.org/item/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/item/AMBP_2014__21_2_1_0/

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