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 (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 : 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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