## Lecture 4. Horizontal Brownian motions on bundles and Hopf fibrations

Let us now turn to some examples of some horizontal Brownian motions associated with submersions.

We come back first to an example studied earlier that encompasses the Heisenberg group. Let
$\alpha=\sum_{i=1}^n \alpha^i(x)dx_i$
be a smooth one-form on $\mathbb{R}^n$ and let $(B_t)_{t \ge 0}$ be a $n$-dimensional Brownian motion. The process

$X_t=\left(B_t, \int_{B[0,t]} \alpha \right)$
is a diffusion process in $\mathbb{R}^n \times \mathbb{R}$ with generator

$L=\frac{1}{2} \sum_{i=1} X_i^2,$
where
$X_i=\frac{\partial}{\partial x_i} +\alpha^i(x) \frac{\partial}{\partial z} .$
We can then interpret $(X_t)_{t \ge 0}$ as a horizontal Brownian motion. Indeed, consider the Riemannian metric $g$ on $\mathbb{R}^n \times \mathbb{R}$ that makes $X_1,\cdots, X_n , \frac{\partial}{\partial z}$ orthonormal. The map

$\pi : (\mathbb{R}^n \times \mathbb{R}, g) \to (\mathbb{R}^n, \mathbf{eucl.})$
such that $\pi (x,z)=x$ is then a Riemannian submersion and $\sum_{i=1} X_i^2$ is the horizontal Laplacian of this submersion. Therefore $(X_t)_{t \ge 0}$ is a horizontal Brownian motion for this submersion.

A second class of examples that naturally arise in stochastic calculus are horizontal Brownian motions on vector bundles. We present here the case of the tangent bundle, but the construction may be extended to any vector bundle. Let $(\mathbb{M},g)$ be a smooth and connected Riemannian manifold with dimension $n$. Let $(B_t)_{t \ge 0}$ be a Brownian motion on $\mathbb M$ started at $x$. Let $\tau_{0,t} : T_x \mathbb M \to T_{B_t} \mathbb{M}$ be the stochastic parallel transport along the paths of $(B_t)_{t \ge 0}$. Let now $v \in T_x \mathbb M$ and consider the tangent bundle $T\mathbb M$ valued process:

$X_t=\left( B_t , \tau_{0,t} v \right).$
The process $(X_t)_{t \ge 0}$ can be interpreted as a horizontal Brownian for some Riemannian submersion. The submersion is simply the bundle projection map $\pi : T\mathbb{M} \to \mathbb{M}$. One then needs to construct a Riemannian metric on $T\mathbb{M}$ that makes $\pi$ a Riemannian submersion. Call a $C^1$ curve $\gamma(t)=(x(t),v(t))$ to be horizontal if $v(t)$ is parallelly transported along $x$. This uniquely determines the rank $n$ horizontal bundle in $TT\mathbb{M}$. Now, if $X$ is a vector field on $\mathbb{M}$, define its horizontal lift $X^h$ as the unique horizontal vector field on $TT\mathbb{M}$ that projects onto $X$. Define its vertical lift as the unique vertical vector field $X^v$ on $TT\mathbb{M}$ such that for every smooth $f:\mathbb{M} \to \mathbb{R}$ one has

$X^v (f^*)=Xf,$
where $f^*(x,v)=df_x (v)$. The Sasaki metric $g_S$ on $T\mathbb{M}$ is then the unique metric such that if $X_1,\cdots, X_n$ is a local orthonormal frame on $\mathbb{M}$, then $X^h_1,\cdots, X^h_n,X_1^v,\cdots,X_n^v$ is a local orthonormal frame on $T\mathbb{M}$. It is then easy to check that $\pi$ is then a Riemannian submersion with totally geodesic fibers and that $(X_t)_{t \ge 0}$ is a horizontal Brownian motion for this submersion.

A similar construction works in the orthonormal frame bundle. Let $(\mathbb{M},g)$ be a smooth and connected Riemannian manifold with dimension $n$ and let $(U_t)_{t \ge 0}$ be the horizontal Brownian motion on the orthonormal frame bundle $O(\mathbb{M})$, that is $(U_t)_{t \ge 0}$ solves the stochastic differential equation

$dU_t=\sum_{i=1}^n H_i(U_t) \circ dB^i_t,$
where $H_1,\cdots,H_n$ are the fundamental horizontal vector fields. Then, similarly as before, one can easily interpret $(U_t)_{t \ge 0}$ as the horizontal Brownian motion of a Riemannian submersion.

The most general construction on bundles is the following. Let $\mathbb{M}$ be a principal bundle over $\mathbb B$ with fiber $\mathbf F$ and structure group $\mathbb G$. Then, given a Riemannian metric $j$ on $\mathbb B$, a $\mathbb G$-invariant metric $k$ on $\mathbf F$ and a $\mathbb G$ connection form $\omega$, there exists a unique Riemannian metric $g$ on $\mathbb M$ such that the bundle projection map $\pi: \mathbb M \to \mathbb B$ is a Riemannian submersion with totally geodesic fibers isometric to $(\mathbf{F},k)$ and such that the horizontal distribution of $\omega$ is the orthogonal complement of the vertical distribution.

We finish the lecture with two canonical examples of horizontal Brownian motions which are related to the Hopf fibrations.

The complex projective space $\mathbb{CP}^n$ can be defined as the set of complex lines in $\mathbb{C}^{n+1}$. To parametrize points in $\mathbb{CP}^n$, it is convenient to use the local inhomogeneous coordinates given by $w_j=z_j/z_{n+1}$, $1 \le j \le n$, $z \in \mathbb{C}^{n+1}$, $z_{n+1}\neq 0$. In these coordinates, the Riemannian structure of $\mathbb{CP}^n$ is easily worked out from the standard Riemannian structure of the Euclidean sphere. Indeed, if we consider the unit sphere

$\mathbb S^{2n+1}=\lbrace z=(z_1,\cdots,z_{n+1})\in \mathbb{C}^{n+1}, \| z \| =1\rbrace,$
then, at each point, the differential of the map $\mathbb S^{2n+1} -\{z_{n+1}=0 \} \to \mathbb{CP}^n$, $(z_1,\cdots,z_{n+1}) \to (z_1/z_{n+1},\cdots,z_n/z_{n+1})$ is an isometry between the orthogonal space of its kernel and the corresponding tangent space to $\mathbb{CP}^n$. This map actually is the local description of a globally defined Riemannian submersion $\mathbb S^{2n+1} \to \mathbb{CP}^n$, that can be constructed as follows. There is an isometric group action of $\mathbb{S}^1=\mathbf{U}(1)$ on $\mathbb S^{2n+1}$ which is defined by

$e^{i\theta}\cdot(z_1,\cdots, z_n) = (e^{i\theta} z_1,\cdots, e^{i\theta} z_n).$

The quotient space $\mathbb S^{2n+1} / \mathbf{U}(1)$ can be identified with $\mathbb{CP}^n$ and the projection map $\pi : \mathbb S^{2n+1} \to \mathbb{CP}^n$ is a Riemannian submersion with totally geodesic fibers isometric to $\mathbf{U}(1)$. The fibration

$\mathbf{U}(1) \to \mathbb S^{2n+1} \to \mathbb{CP}^n$
is called the Hopf fibration.

The submersion $\pi$ allows to construct the Brownian motion on $\mathbb{CP}^n$ from the Brownian motion on $\mathbb S^{2n+1}$. Indeed, let $(z(t))_{t \ge 0}$ be a Brownian motion on $\mathbb S^{2n+1}$ started at the north pole. Since $\mathbb{P}( \exists t \ge 0, z_{n+1}(t)=0 )=0$, one can use the local description of the submersion $\pi$ to infer that

$w(t)= \left( \frac{z_1(t)}{z_{n+1}(t)} , \cdots, \frac{z_n(t)}{z_{n+1}(t)}\right), \quad t \ge 0,$
is a Brownian motion on $\mathbb{CP}^n$.

Consider now the one-form $\alpha$ on $\mathbb{CP}^n$ which is the pushforward by $\pi$ of the standard contact form of $\mathbb S^{2n+1}$. In local inhomogeneous coordinates, we have

$\alpha=\frac{i}{2(1+|w|^2)}\sum_{j=1}^n(w_jd\overline{w_j}-\overline{w_j}dw_j)$
where $|w|^2=\sum_{j=1}^n |w_j|^2$. It is easy to compute that

$d\alpha=\frac{i}{(1+|w|^2)^2}\left((1+|w|^2)\sum_{j=1}^ndw_j\wedge d\overline{w}_j-\sum_{j,k=1}^n\overline{w}_jw_k dw_j\wedge d\overline{w}_k \right).$
Thus $d\alpha$ is almost everywhere the Kahler form that induces the standard Fubini-Study metric on $\mathbb{CP}^n$. The following definition is therefore natural:

Definition: Let $(w(t))_{t \ge 0}$ be a Brownian motion on $\mathbb{CP}^n$ started at 0. The generalized stochastic area process of $(w(t))_{t \ge 0}$ is defined by
$\theta(t)=\int_{w[0,t]} \alpha=\frac{i}{2}\sum_{j=1}^n \int_0^t \frac{w_j(s) d\overline{w_j}(s)-\overline{w_j}(s) dw_j(s)}{1+|w(s)|^2},$
where the above stochastic integrals are understood in the Stratonovitch, or equivalently in the Ito sense.

We have then the following representation for the horizontal Brownian motion of the submersion $\pi$.

Theorem: Let $(w(t))_{t \ge 0}$ be a Brownian motion on $\mathbb{CP}^n$ started at 0 and $(\theta(t))_{t\ge 0}$ be its stochastic area process. The $\mathbb{S}^{2n+1}$-valued diffusion process
$X_t=\frac{e^{-i\theta(t)} }{\sqrt{1+|w(t)|^2}} \left( w(t),1\right), \quad t \ge 0$
is a horizontal Brownian motion for the submersion $\pi$.

A similar construction works on the complex hyperbolic space. As a set, the complex hyperbolic space $\mathbb{CH}^n$ can be defined as the open unit ball in $\mathbb{C}^n$. Its Riemannian structure can be constructed as follows. Let

$\mathbb H^{2n+1}=\{ z \in \mathbb{C}^{n+1}, | z_1|^2+\cdots+|z_n|^2 -|z_{n+1}|^2=-1 \}$
be the $2n+1$ dimensional anti-de Sitter space. We endow $\mathbb H^{2n+1}$ with its standard Lorentz metric with signature $(2n,1)$. The Riemannian structure on $\mathbb{CH}^n$ is then such that the map
$\begin{array}{llll} \pi :& \mathbb H^{2n+1} & \to & \mathbb{CH}^n \\ & (z_1,\cdots,z_{n+1}) & \to & \left( \frac{z_1}{z_{n+1}}, \cdots, \frac{z_n}{z_{n+1}}\right) \end{array}$
is an indefinite Riemannian submersion whose one-dimensional fibers are definite negative. This submersion is associated with a fibration. Indeed, the group $\mathbf{U}(1)$ acts isometrically on $\mathbb H^{2n+1}$, and the quotient space of $\mathbb H^{2n+1}$ by this action is isometric to $\mathbb{CH}^n$. The fibration

$\mathbf{U}(1)\to\mathbb H^{2n+1}\to\mathbb{CH}^n$
is called the anti-de Sitter fibration.

To parametrize $\mathbb{CH}^n$, we will use the global inhomogeneous coordinates given by $w_j=z_j/z_{n+1}$ where $(z_1,\dots, z_n)\in M$ with $M=\{z\in \mathbb{C}^{n,1}, \sum_{k=1}^n|z_{k}|^2-|z_{n+1}|^2<0 \}$. Let $\alpha$ be the one-form on $\mathbb{CH}^n$ which is the push-forward by the submersion $\pi$ of the standard contact form on $\mathbb H^{2n+1}$. In inhomogeneous coordinates, we compute

$\alpha=\frac{i}{2(1-|w|^2)}\sum_{j=1}^n(w_jd\overline{w_j}-\overline{w_j}dw_j),$
where $|w|^2=\sum_{j=1}^n|w_j|^2<1$. A simple computation yields

$d\alpha=\frac{i}{(1-|w|^2)^2}\left((1-|w|^2)\sum_{j=1}^ndw_j\wedge d\overline{w}_j-\sum_{j,k=1}^n\overline{w}_jw_k dw_j\wedge d\overline{w}_k \right).$
Thus $d\alpha$ is exactly the Kahler form which induces the standard Bergman metric on $\mathbb{CH}^n$. We can then naturally define the stochastic area process on $\mathbb{CH}^n$ as follows:
Definition:  Let $(w(t))_{t \ge 0}$ be a Brownian motion on $\mathbb{CH}^n$ started at 0. The generalized stochastic area process of $(w(t))_{t \ge 0}$ is defined by

$\theta(t)=\int_{w[0,t]} \alpha=\frac{i}{2}\sum_{j=1}^n \int_0^t \frac{w_j(s) d\overline{w_j}(s)-\overline{w_j}(s) dw_j(s)}{1-|w(s)|^2},$
where the above stochastic integrals are understood in the Stratonovitch sense or equivalently Ito sense.

As in in the Heisenberg group case or the Hopf fibration case, the stochastic area process is intimately related to the horizontal Brownian motion on the total space of the fibration.

Theorem: Let $(w(t))_{t \ge 0}$ be a Brownian motion on $\mathbb{CH}^n$ started at 0 and $(\theta(t))_{t\ge 0}$ be its stochastic area process. The $\mathbb H^{2n+1}$-valued diffusion process

$Y_t=\frac{e^{i\theta_t} }{\sqrt{1-|w(t)|^2}} \left( w(t),1\right), \quad t \ge 0$
is the horizontal lift at $(0,1)$ of $(w(t))_{t \ge 0}$ by the submersion $\pi$.

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