Stochastic areas, winding numbers and Hopf fibrations

@article{Baudoin2016StochasticAW,
  title={Stochastic areas, winding numbers and Hopf fibrations},
  author={Fabrice Baudoin and Jing Wang},
  journal={Probability Theory and Related Fields},
  year={2016},
  volume={169},
  pages={977-1005}
}
We define and study stochastic areas processes associated with Brownian motions on the complex symmetric spaces $$\mathbb {CP}^n$$CPn and $$\mathbb {CH}^n$$CHn. The characteristic functions of those processes are computed and limit theorems are obtained. In the case $$n=1$$n=1, we also study windings of the Brownian motion on those spaces and compute the limit distributions. For $$\mathbb {CP}^n$$CPn the geometry of the Hopf fibration plays a central role, whereas for $$\mathbb {CH}^n$$CHn it… 
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References

SHOWING 1-10 OF 31 REFERENCES
The subelliptic heat kernel on the CR sphere
We study the heat kernel of the sub-Laplacian $$L$$ on the CR sphere $$\mathbb{S }^{2n+1}$$. An explicit and geometrically meaningful formula for the heat kernel is obtained. As a by-product we
Limits of random differential equations on manifolds
TLDR
Under Hörmander type conditions on L0, it is proved that the stochastic processes y_{t\over \epsilon }^(ϵ approaches zero) converge weakly and in the Wasserstein topologies.
The Subelliptic Heat Kernel on the Anti-de Sitter Space
We study the subelliptic heat kernel of the sub-Laplacian on a 2n+1-dimensional anti-de Sitter space ℍ2n+1 which also appears as a model space of a CR Sasakian manifold with constant negative
On hyperbolic Bessel processes and beyond
We investigate distributions of hyperbolic Bessel processes. We find links between the hyperbolic cosinus of the hyperbolic Bessel processes and the functionals of geometric Brownian motion. We
The Subelliptic Heat Kernels on SL(2, ℝ) and on its Universal Covering $\widetilde{\mathbf{SL}(2,\mathbb{R})}$: Integral Representations and Some Functional Inequalities
In this paper, we study a subelliptic heat kernel on the Lie group SL(2, ℝ) and on its universal covering $\widetilde{\mathbf{SL}(2,\mathbb{R})}$. The subelliptic structure on SL(2,ℝ) comes from the
Probability laws related to the Jacobi theta and Riemann zeta functions, and Brownian excursions
This paper reviews known results which connect Riemann’s integral representations of his zeta function, involving Jacobi’s theta function and its derivatives, to some particular probability laws
Areas of planar Brownian curves
Addresses the problem of the algebraic area enclosed by a Brownian curve in two dimensions, recently reconsidered by Khandekar and Wiegel (1988). The author recalls the principal results actually
The Subelliptic Heat Kernels of the Quaternionic Hopf Fibration
The main goal of this work is to study the sub-Laplacian of the unit sphere which is obtained by lifting with respect to the Hopf fibration the Laplacian of the quaternionic projective space. We
Continuous martingales and Brownian motion
0. Preliminaries.- I. Introduction.- II. Martingales.- III. Markov Processes.- IV. Stochastic Integration.- V. Representation of Martingales.- VI. Local Times.- VII. Generators and Time Reversal.-
...
1
2
3
4
...