We now study in more details the geometric structure behind the diffusion underlying the Levy area process

where , , is a two dimensional Brownian motion started at 0. Let us recall that if we consider the 3-dimensional process

then is a Markov process with generator

where are the following vector fields

Denote the vector field

We have then Lie brackets commutation relations

As a consequence, generate a 3-dimensional nilpotent Lie algebra of (complete) vector fields. This is the Lie algebra of the Heisenberg group

where the non-commutative group law is given by

For , one has

Observe now that

Therefore, is independent from and distributed as . It is therefore natural to call a Brownian motion in the Heisenberg group. Observe that form a basis of the Lie algebra, but that the generator of only involves and . Thus, the *direction* is missing. Calling the set of horizontal directions, we then refer to as a **horizontal Brownian motion**.

This construction is easily extended to higher dimensions.

Let be a Brownian motion in started at 0. This means that and are two independent Brownian motions in . We can then consider the one-form

and the generalized Levy area

The process

is then a diffusion process in with generator

where

being the last coordinate in . If we denote , we have the following Lie brackets relations

and

In particular, the Hormander’s condition is satisfied. We also see that generate the Lie algebra of the -dimensional Heisenberg group

where the group law is given by

As before, we can then interpret as a horizontal Brownian motion on .

The group structure is specific to the particular choice of the one-form . If one wants to study more general situations, one has to use some Riemannian geometry.

Consider for instance a general smooth one-form

on and let be a -dimensional Brownian motion. We have

where the stochastic integrals have to be understood in the Stratonovitch sense. The process

is a diffusion process in with generator

where

We have

Therefore, if the two-form is never 0, then the Hormander’s condition is satisfied.

We would like to call the horizontal Brownian motion of some relevant geometric structure…