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

Nizar Demni

doi:10.5802/ambp.339

Nizar Demni ¹

¹ Institut de Recherche en Mathématiques de Rennes Université de Rennes 1 Campus de Beaulieu 35042 Rennes FRANCE

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

Résumé

Let ${(U_{t})}_{t \geq 0}$ 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 $(| U_{t}^{1} |^{2}, | U_{t}^{2} |^{2})$ , and express them through Jacobi polynomials in the simplices of $ℝ$ and $ℝ^{2}$ respectively. More generally, the distribution of $(| U_{t}^{1} |^{2}, \dots, | U_{t}^{k} |^{2}), 2 \leq k \leq N - 1$ 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 $1 \leq k \leq N - 2$ , integrations by parts performed on the pde lead to a heat equation in the simplex of $ℝ^{k}$ .

DOI : 10.5802/ambp.339

Mots clés : Brownian motion, complex projective space, Dirichlet distribution, Jacobi polynomials in the simplex

Affiliations des auteurs :

Nizar Demni ¹

¹ Institut de Recherche en Mathématiques de Rennes Université de Rennes 1 Campus de Beaulieu 35042 Rennes FRANCE

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Nizar Demni. 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. https://ambp.centre-mersenne.org/articles/10.5802/ambp.339/

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