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

Demni, Nizar. "Distributions of truncations of the heat kernel on the complex projective space." Annales mathématiques Blaise Pascal 21.2 (2014): 1-20. <http://eudml.org/doc/275449>.

@article{Demni2014,
abstract = {Let $(U_t)_\{t \ge 0\}$ be a Brownian motion valued in the complex projective space $\mathbb\{C\}P^\{N-1\}$. 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 $\mathbb\{R\}$ and $\mathbb\{R\}^2$ respectively. More generally, the distribution of $(|U_t^\{1\}|^2, \dots , |U_t^k|^2), \;2 \le k \le N-1$ may be derived using the decomposition of the unitary spherical harmonics under the action of the unitary group $\mathcal\{U\}(N-k+1)$ 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 $\mathbb\{R\}^k$.},
affiliation = {Institut de Recherche en Mathématiques de Rennes Université de Rennes 1 Campus de Beaulieu 35042 Rennes FRANCE},
author = {Demni, Nizar},
journal = {Annales mathématiques Blaise Pascal},
keywords = {Brownian motion; complex projective space; Dirichlet distribution; Jacobi polynomials in the simplex; unitary spherical harmonics; Jacobi polynomials; simplices; Laplace transform; heat equation},
language = {eng},
month = {7},
number = {2},
pages = {1-20},
publisher = {Annales mathématiques Blaise Pascal},
title = {Distributions of truncations of the heat kernel on the complex projective space},
url = {http://eudml.org/doc/275449},
volume = {21},
year = {2014},
}

TY - JOUR
AU - Demni, Nizar
TI - Distributions of truncations of the heat kernel on the complex projective space
JO - Annales mathématiques Blaise Pascal
DA - 2014/7//
PB - Annales mathématiques Blaise Pascal
VL - 21
IS - 2
SP - 1
EP - 20
AB - Let $(U_t)_{t \ge 0}$ be a Brownian motion valued in the complex projective space $\mathbb{C}P^{N-1}$. 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 $\mathbb{R}$ and $\mathbb{R}^2$ respectively. More generally, the distribution of $(|U_t^{1}|^2, \dots , |U_t^k|^2), \;2 \le k \le N-1$ may be derived using the decomposition of the unitary spherical harmonics under the action of the unitary group $\mathcal{U}(N-k+1)$ 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 $\mathbb{R}^k$.
LA - eng
KW - Brownian motion; complex projective space; Dirichlet distribution; Jacobi polynomials in the simplex; unitary spherical harmonics; Jacobi polynomials; simplices; Laplace transform; heat equation
UR - http://eudml.org/doc/275449
ER -