Densities of generalized stochastic areas and windings arising from Anti-de Sitter and Hopf fibrations

In the first part of this paper, we derive explicit expressions of the semi-group densities of generalized stochastic areas arising from the Anti-de Sitter and the Hopf fibrations. Motivated by the number-theoretical connection between the Heisenberg group and Dirichlet series, we express the Mellin transform of the generalized stochastic area corresponding to the one-dimensional Anti de Sitter fibration as a series of Riemann Zeta function evaluated at integers. In the second part of the paper, we derive fixed-time marginal densities of winding processes around the origin in the Poincaré disc and in the complex projective line.