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

Let ${\left(U_t\right)}_{t\ge 0}$ be a Brownian motion valued in the complex projective space $ℂP^{N-1}$. Using unitary spherical harmonics of homogeneous degree zero, we derive the densities of $|U_t^1{|}^2$ and of $\left(\right|U_t^1{|}^2,\left|U_t^2{|}^2\right)$, and express them through Jacobi polynomials in the simplices of $ℝ$ and ${ℝ}^2$ respectively. More generally, the distribution of $\left(\right|U_t^1{|}^2,\cdots ,\left|U_t^k{|}^2\right),2\le k\le N-1$ 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 $1\le k\le N-2$, integrations by parts performed on the pde lead to a heat equation in the simplex of ${ℝ}^k$.