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

doi:10.1016/j.indag.2019.12.005

Indagationes Mathematicae

Volume 31, Issue 2, March 2020, Pages 204-222

https://doi.org/10.1016/j.indag.2019.12.005 Get rights and content

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 derive fixed-time marginal densities of winding processes around the origin in the Poincaré disc and in the complex projective line.

Section snippets

Motivation: The Heisenberg group case

The Lévy stochastic area $A_{t} ≔ \int_{0}^{t} B_{s}^{1} d B_{s}^{2} - B_{s}^{2} d B_{s}^{1}, t \geq 0,$ where $B ≔ (B^{1}, B^{2})$ is a planar Brownian motion, is a very interesting object in both probability theory and mathematical physics [7]. It arises naturally from the heat kernel of the three-dimensional Heisenberg group $H_{3} = ℂ \times R$ since the latter is endowed with the standard contact form written in local coordinates $(x, y, t)$ : $η_{H} ≔ d t + (x d y - y d x) .$ This form is actually the pull-back of the standard Kähler form $α_{H} ≔ x d y - y d x$ on $ℂ$ with respect to the fibration: $π : H_{3} \to$

Special functions

We start with the Gamma function defined for $x > 0$ by: $Γ (x) = \int_{0}^{\infty} e^{- u} u^{x - 1} d u .$ This function satisfies the Legendre duplication formula: $\sqrt{π} Γ (2 x + 1) = 2^{2 x} Γ (x + \frac{1}{2}) Γ (x + 1) .$ Next, let $k \geq 1$ be a non negative integer. Then the Pochhammer symbol is defined by: ${(x)}_{k} = (x + k - 1) \dots (x + 1) x, x \in R,$ with the convention ${(x)}_{0} ≔ 1$ . When $x > 0$ , we can express it through the Gamma function as: ${(x)}_{k} = \frac{Γ (x + k)}{Γ (x)},$ while ${(- n)}_{k} = \frac{{(- 1)}^{k} n!}{(n - k)!}, if k \leq n, = 0, if k > n .$ Now, the hypergeometric series $_{p} F_{q}$ is defined by: $_{p} F_{q} (a_{1}, \dots, a_{p}; b_{1}, \dots; b_{q}; x) ≔ \sum_{k \geq 0} \frac{{(a_{1})}_{k} \dots {(a_{p})}_{k}}{{(b_{1})}_{q} \dots (}$

The Anti-de Sitter fibration

The Anti de Sitter space is the hypersurface in $ℂ^{n + 1}$ defined by: ${AdS}_{n} ≔ {(z_{1}, \dots, z_{n + 1}) \in ℂ^{n + 1}, {| z_{1} |}^{2} + {| z_{2} |}^{2} + \dots + {| z_{n} |}^{2} - {| z_{n + 1} |}^{2} = - 1} .$ It inherits from $R^{2 n, 2}$ a Lorentzian $(2 n, 1)$ -metric of constant negative curvature and the circle acts on it in a natural way. The coset space of this action is isometric to the complex hyperbolic ball $ℂ H^{n}$ and the projection map ${AdS}_{n} \to ℂ H^{n},$ is indeed a fibration. In the chart ${z_{n + 1} \neq 0}$ , this fibration sends the coordinate $z_{j}$ to $w_{j} = z_{j} ∕ z_{n + 1}$ giving rise to inhomogeneous coordinates $(w_{1},$

The Hopf fibration

In this section, we deal with the spherical analogue the AdS fibration, commonly known as the Hopf fibration [9]. Here, the base space is the complex projectif space ${CP}^{n} \subset ℂ^{n + 1}$ and the total space is the odd-dimensional sphere: $S^{2 n + 1} ≔ {(z_{1}, \dots, z_{n + 1}) \in ℂ^{n + 1}, {| z_{1} |}^{2} + {| z_{2} |}^{2} + \dots + {| z_{n} |}^{2} + {| z_{n + 1} |}^{2} = 1},$ on which the circle acts isometrically. We similarly denote $(w_{1}, \dots, w_{n})$ the inhomogeneous coordinates in the chart $z_{n + 1} \neq 0$ and consider a Brownian motion ${(w (t))}_{t \geq 0}$ on ${CP}^{n}$ starting at zero. Then, the generalized stochastic

Winding processes in $ℂ H^{1}$ and in $ℂ P^{1}$

In this section, we are interested in winding processes around the origin in the Poincaré disc $ℂ H^{1}$ and in the complex projective line $ℂ P^{1}$ . As in the Euclidean setting, these processes are naturally defined as angular parts of Laplace operators of their underlying geometrical models. In [3], the characteristic functions of their fixed-time marginal distributions were expressed as expectations with respect to the hyperbolic Jacobi and the ultraspherical operators respectively (see below), and

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View full text

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

Abstract

Section snippets

Motivation: The Heisenberg group case

Special functions

The Anti-de Sitter fibration

The Hopf fibration

Winding processes in ℂH1 and in ℂP1

Special Functions

Selberg trace formulae for heat and wave kernels of Maass Laplacians on compact forms of the complex hyperbolic space Hn(ℂ),n≥2

Differential Geom. Appl.

J. Wang Stochastic areas, winding numbers and Hopf fibrations

Probab. Theory Related Fields

Integral representation of the sub-elliptic heat kernel on the Anti-de Sitter space

Arch. der Math. (Basel)

Probability laws related to the Jacobi theta and Riemann zeta functions, and Brownian excursions

Bull. Amer. Math. Soc. (N.S.)

The heat semigroup for the Jacobi?Dunkl operator and the related Markov processes

Potential Anal.

Winding processes in $ℂ H^{1}$ and in $ℂ P^{1}$

Selberg trace formulae for heat and wave kernels of Maass Laplacians on compact forms of the complex hyperbolic space $H_{n} (ℂ), n \geq 2$