Generalized Stochastic areas and windings arising from Anti-de Sitter and Hopf fibrations

TL;DR

This paper derives explicit formulas for stochastic areas related to Anti-de Sitter and Hopf fibrations, explores their connections to number theory, and analyzes winding processes in geometric settings.

Contribution

It provides new explicit semi-group densities, links stochastic areas to Riemann Zeta functions, and characterizes winding processes in complex geometries.

Findings

Explicit semi-group densities for generalized stochastic areas.

Mellin transform expressed as Riemann Zeta series.

Fixed-time density and characteristic function expansions for winding processes.

Abstract

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 focus on winding processes around the origin in the Poincar\'e disc and in the complex projective line. More pricesely, we derive the fixed-time marginal density of the former process while we give a ultraspherical series expansion of the characteristic function of the latter.