Let $(U_t)_{ t \geq 0}$ 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 $(|U_t^{ 1}|^2, \dots, |U_t^k|^2), 2 \leq k \leq N-1$ 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 \cite{ Nec-Pel} 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 derived using spherical harmonics. For general $1 \leq k \leq N-2$, the integrations by parts performed on the pde lead to a heat equation in the simplex of $\mathbb{ R}^k$.
Categories: math.PR