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On the law of the index of Brownian loops related to the Hopf and anti-de Sitter fibrations

Fabrice Baudoin, Teije Kuijper

Abstract

We give explicit formulas and asymptotics for the distribution of the index of the Brownian loop in the following geometrical settings: the complex projective line from which two points have been removed; the complex hyperbolic line from which one point has been removed; the odd dimensional spheres from which a great hypersphere has been removed; and the complex anti-de Sitter spaces. Our analysis is based on the geometry of the Hopf and anti-de Sitter fibrations, and on the relationship between winding and area forms.

On the law of the index of Brownian loops related to the Hopf and anti-de Sitter fibrations

Abstract

We give explicit formulas and asymptotics for the distribution of the index of the Brownian loop in the following geometrical settings: the complex projective line from which two points have been removed; the complex hyperbolic line from which one point has been removed; the odd dimensional spheres from which a great hypersphere has been removed; and the complex anti-de Sitter spaces. Our analysis is based on the geometry of the Hopf and anti-de Sitter fibrations, and on the relationship between winding and area forms.

Paper Structure

This paper contains 14 sections, 19 theorems, 168 equations.

Table of Contents

  1. Introduction
  2. Windings on the Hopf fibration
  3. Preliminaries
  4. Jacobi diffusions
  5. Brownian motion on Cpn
  6. Windings on CP1
  7. Windings on S(2n+1)
  8. Windings on the anti-de Sitter fibration
  9. Preliminaries
  10. Hyperbolic Jacobi diffusions
  11. Brownian motion on CHn
  12. Windings on CH1
  13. Windings on the anti de-Sitter space
  14. The index of the Brownian loop on SL(2, R)

Key Result

Theorem 1.1

Let $(X(t))_{0 \le t \le L}$ be a Brownian loop of length $L$ in $\mathbf{SL}(2,\mathbb R)$, i.e. a Brownian motion started from the identity and conditioned to come back to identity at time $L$. Then, for every $k \in \mathbb Z$, where $C >0$ is the normalization constant. In particular, in distribution, when $L\to +\infty$, where $\mathcal{N}(0,1)$ is a normal random variable with mean zero an

Theorems & Definitions (42)

  • Theorem 1.1
  • Definition 2.1
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • Theorem 2.4
  • proof
  • Corollary 2.5
  • proof
  • ...and 32 more