Stochastic areas, winding numbers and Hopf fibrations

Abstract

We define and study stochastic areas processes associated with Brownian motions on the complex symmetric spaces $\mathbb {CP}^n$ and $\mathbb {CH}^n$. The characteristic functions of those processes are computed and limit theorems are obtained. In the case $n=1$, we also study windings of the Brownian motion on those spaces and compute the limit distributions. For $\mathbb {CP}^n$ the geometry of the Hopf fibration plays a central role, whereas for $\mathbb {CH}^n$ it is the anti-de Sitter fibration.

Appendix: Jacobi diffusions

We collect here some well-known facts about Jacobi diffusions (see for instance [10] and the references therein for further details). The Jacobi diffusion is the diffusion on $[0,\pi /2]$ with generator

$$\begin{aligned} {\mathcal {L}}^{\alpha ,\beta }=\frac{1}{2} \frac{\partial ^2}{\partial r^2}+\left( \left( \alpha +\frac{1}{2}\right) \cot r-\left( \beta +\frac{1}{2}\right) \tan r\right) \frac{\partial }{\partial r}, \quad \alpha ,\beta >-1 \end{aligned}$$

defined up to the first time it hits the boundary $\{ 0, \pi /2 \}$.

The point 0 is:

• A regular point for $-1<\alpha <0$;

• An entrance point for $\alpha \ge 0$.

Similarly, the point $\pi /2$ is:

• A regular point for $-1<\beta <0$;

• An entrance point for $\beta \ge 0$.

If r is a Jacobi diffusion with generator ${\mathcal {L}}^{\alpha ,\beta }$, then it is easily seen that $\rho =\cos 2r$ is a diffusion with generator $2{\mathcal {G}}^{\alpha ,\beta }$ where,

$$\begin{aligned} {\mathcal {G}}^{\alpha ,\beta }=(1-\rho ^2)\frac{\partial ^2}{\partial \rho ^2}-\left( (\alpha +\beta +2)\rho +\alpha -\beta \right) \frac{\partial }{\partial \rho } \end{aligned}$$

(4.2)

The spectrum and eigenfunctions of ${\mathcal {G}}^{\alpha ,\beta }$ are known. Let us denote by $P_m^{\alpha ,\beta }(x)$, $m\in {\mathbb {Z}}_{\ge 0}$ the Jacobi polynomials given by

$$\begin{aligned} P_m^{\alpha ,\beta }(x)=\frac{(-1)^m}{2^mm!(1-x)^{\alpha }(1+x)^\beta }\frac{d^m}{dx^m}((1-x)^{\alpha +m}(1+x)^{\beta +m}). \end{aligned}$$

It is known that $\{P_m^{\alpha ,\beta }(x)\}$ is orthonormal in $L^2([-1,1], 2^{-\alpha -\beta -1}(1+x)^{\beta }(1-x)^{\alpha }dx)$ and satisfies

$$\begin{aligned} {\mathcal {G}}^{\alpha ,\beta }P_m^{\alpha ,\beta }(x)=-m(m+\alpha +\beta +1)P_m^{\alpha ,\beta }(x). \end{aligned}$$

If we denote by $p^{\alpha ,\beta }_t(x,y)$ transition density with respect to the Lebesgue measure of the diffusion $\rho$ starting from $x \in (-1,1)$, then we have

$$\begin{aligned} p^{\alpha ,\beta }_t(x,y)&=2^{-\alpha -\beta -1}(1+y)^{\beta }(1-y)^{\alpha }\sum _{m=0}^{+\infty } (2m+\alpha +\beta +1)\\ {}&\qquad \times \frac{\Gamma (m+\alpha +\beta +1)\Gamma (m+1)}{\Gamma (m+\alpha +1)\Gamma (m+\beta +1)} e^{-2m(m+\alpha +\beta +1)t}P_m^{\alpha ,\beta }(x)P_m^{\alpha ,\beta }(y). \end{aligned}$$

In particular, when 1 is an entrance point, that is $\alpha \ge 0$, we obtain

$$\begin{aligned} p^{\alpha ,\beta }_t(1,y)&=2^{-\alpha -\beta -1}(1+y)^{\beta }(1-y)^{\alpha } \sum _{m=0}^{+\infty } (2m+\alpha +\beta +1)\\&\qquad \times \frac{\Gamma (m+\alpha +\beta +1)}{\Gamma (m+\beta +1)\Gamma (\alpha +1)} e^{-2m(m+\alpha +\beta +1)t}P_m^{\alpha ,\beta }(y). \end{aligned}$$

By denoting $q_t^{\alpha ,\beta }$ the transition density of r, we obtain then for $\alpha ,\beta \ge 0$,

$$\begin{aligned} q^{\alpha ,\beta }_t(r_0,r)&=2(\cos r)^{2\beta +1} (\sin r)^{2\alpha +1}\\&\qquad \cdot \sum _{m=0}^{+\infty } (2m+\alpha +\beta +1)\\&\qquad \times \frac{\Gamma (m+\alpha +\beta +1)\Gamma (m+1)}{\Gamma (m+\alpha +1)\Gamma (m+\beta +1)}\\&\quad e^{-2m(m+\alpha +\beta +1)t}P_m^{\alpha ,\beta }(\cos 2r_0)P_m^{\alpha ,\beta }(\cos 2r). \end{aligned}$$

and

$$\begin{aligned} q^{\alpha ,\beta }_t(0,r)&=2(\cos r)^{2\beta +1} (\sin r)^{2\alpha +1} \sum _{m=0}^{+\infty } (2m+\alpha +\beta +1)\\&\qquad \times \frac{\Gamma (m+\alpha +\beta +1)}{\Gamma (m+\beta +1)\Gamma (\alpha +1)} e^{-2m(m+\alpha +\beta +1)t}P_m^{\alpha ,\beta }(\cos 2r). \end{aligned}$$