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
is then a diffusion process in with generator
being the last coordinate in . If we denote , we have the following Lie brackets relations
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
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…