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

be a smooth one-form on and let be a -dimensional Brownian motion. The process

is a diffusion process in with generator

where

We can then interpret as a horizontal Brownian motion. Indeed, consider the Riemannian metric on that makes orthonormal. The map

such that is then a Riemannian submersion and is the horizontal Laplacian of this submersion. Therefore 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 be a smooth and connected Riemannian manifold with dimension . Let be a Brownian motion on started at . Let be the stochastic parallel transport along the paths of . Let now and consider the tangent bundle valued process:

The process can be interpreted as a horizontal Brownian for some Riemannian submersion. The submersion is simply the bundle projection map . One then needs to construct a Riemannian metric on that makes a Riemannian submersion. Call a curve to be horizontal if is parallelly transported along . This uniquely determines the rank horizontal bundle in . Now, if is a vector field on , define its horizontal lift as the unique horizontal vector field on that projects onto . Define its vertical lift as the unique vertical vector field on such that for every smooth one has

where . The **Sasaki metric** on is then the unique metric such that if is a local orthonormal frame on , then is a local orthonormal frame on . It is then easy to check that is then a Riemannian submersion with totally geodesic fibers and that is a horizontal Brownian motion for this submersion.

A similar construction works in the orthonormal frame bundle. Let be a smooth and connected Riemannian manifold with dimension and let be the horizontal Brownian motion on the orthonormal frame bundle , that is solves the stochastic differential equation

where are the fundamental horizontal vector fields. Then, similarly as before, one can easily interpret as the horizontal Brownian motion of a Riemannian submersion.

The most general construction on bundles is the following. Let be a principal bundle over with fiber and structure group . Then, given a Riemannian metric on , a -invariant metric on and a connection form , there exists a unique Riemannian metric on such that the bundle projection map is a Riemannian submersion with totally geodesic fibers isometric to and such that the horizontal distribution of 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 can be defined as the set of complex lines in . To parametrize points in , it is convenient to use the local inhomogeneous coordinates given by , , , . In these coordinates, the Riemannian structure of is easily worked out from the standard Riemannian structure of the Euclidean sphere. Indeed, if we consider the unit sphere

then, at each point, the differential of the map , is an isometry between the orthogonal space of its kernel and the corresponding tangent space to . This map actually is the local description of a globally defined Riemannian submersion , that can be constructed as follows. There is an isometric group action of on which is defined by

The quotient space can be identified with and the projection map is a Riemannian submersion with totally geodesic fibers isometric to . The fibration

is called the Hopf fibration.

The submersion allows to construct the Brownian motion on from the Brownian motion on . Indeed, let be a Brownian motion on started at the north pole. Since , one can use the local description of the submersion to infer that

is a Brownian motion on .

Consider now the one-form on which is the pushforward by of the standard contact form of . In local inhomogeneous coordinates, we have

where . It is easy to compute that

Thus is almost everywhere the Kahler form that induces the standard Fubini-Study metric on . The following definition is therefore natural:

**Definition: **Let be a Brownian motion on started at 0. The generalized stochastic area process of is defined by

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 .

**Theorem: **Let be a Brownian motion on started at 0 and be its stochastic area process. The -valued diffusion process

is a horizontal Brownian motion for the submersion .

A similar construction works on the complex hyperbolic space. As a set, the complex hyperbolic space can be defined as the open unit ball in . Its Riemannian structure can be constructed as follows. Let

be the dimensional anti-de Sitter space. We endow with its standard Lorentz metric with signature . The Riemannian structure on is then such that the map

is an indefinite Riemannian submersion whose one-dimensional fibers are definite negative. This submersion is associated with a fibration. Indeed, the group acts isometrically on , and the quotient space of by this action is isometric to . The fibration

is called the anti-de Sitter fibration.

To parametrize , we will use the global inhomogeneous coordinates given by where with . Let be the one-form on which is the push-forward by the submersion of the standard contact form on . In inhomogeneous coordinates, we compute

where . A simple computation yields

Thus is exactly the Kahler form which induces the standard Bergman metric on . We can then naturally define the stochastic area process on as follows:

**Definition: **Let be a Brownian motion on started at 0. The generalized stochastic area process of is defined by

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 be a Brownian motion on started at 0 and be its stochastic area process. The -valued diffusion process

is the horizontal lift at of by the submersion .