Distributions of truncations of the heat kernel on the complex projective space
Annales mathématiques Blaise Pascal, Volume 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
Keywords: Brownian motion, complex projective space, Dirichlet distribution, Jacobi polynomials in the simplex
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, Volume 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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