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

Demni, Nizar

Annales mathématiques Blaise Pascal, Tome 21 (2014), p. 1-20 / Harvested from Numdam

Résumé

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 $\mathbb{R}^k$.

Détail

Publié le : 2014-01-01

MR 3322612

DOI : https://doi.org/10.5802/ambp.339

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