Lecture 2. Horizontal Brownian motion on the Heisenberg group

We now study in more details the geometric structure behind the diffusion underlying the Levy area process
$S_t=\int_0^t B^1_s dB^2_s-B^2_s dB^1_s,$
where $B_t=(B^1_t,B^2_t)$, $t \ge 0$, is a two dimensional Brownian motion started at 0. Let us recall that if we consider the 3-dimensional process

$X_t=(B^1_t,B^2_t,S_t),$
then $X_t$ is a Markov process with generator

$L=\frac{1}{2}( X^2+Y^2)$
where $X,Y$ are the following vector fields

$X=\frac{\partial}{\partial x}-y \frac{\partial}{\partial z}$

$Y=\frac{\partial}{\partial y}+x \frac{\partial}{\partial z}.$
Denote $Z$ the vector field

$Z=\frac{\partial}{\partial z}$
We have then Lie brackets commutation relations
$[X,Y]=2Z, [X,Z]=[Y,Z]=0.$
As a consequence, $X,Y,Z$ generate a 3-dimensional nilpotent Lie algebra of (complete) vector fields. This is the Lie algebra of the Heisenberg group

$\mathbb{H}^3=\{ (x,y,z) \in \mathbb{R}^3 \}$
where the non-commutative group law is given by
$(x_1,y_1,z_1) \star (x_2,y_2,z_2) =(x_1+x_2,y_2+y_2,z_1+z_2+x_1y_2-x_2y_1).$

For $s \le t$, one has

$X_s^{-1} \star X_t =(-B^1_s,-B^2_s,-S_s) \star (B^1_t,B^2_t,S_t) =(B^1_t-B^1_s,B^2_t-B^2_s,S_t-S_s-B^1_sB^2_t+B^2_sB^1_t)$
Observe now that

$S_t-S_s-B^1_sB^2_t+B^2_sB^1_t=\int_s^t (B^1_u-B^1_s)dB^2_u-(B^2_u-B^2_s)dB^1_u.$
Therefore, $X_s^{-1} \star X_t$ is independent from $\sigma(X_u, u \le s)$ and distributed as $X_{t-s}$. It is therefore natural to call $(X_t)_{t\ge 0}$ a Brownian motion in the Heisenberg group. Observe that $X,Y,Z$ form a basis of the Lie algebra, but that the generator of  $X_t$ only involves $X$ and $Y$. Thus, the direction $Z$ is missing. Calling $\mathbf{span}(X,Y)$ the set of horizontal directions, we then refer to $X_t$ as a horizontal Brownian motion.

This construction is easily extended to higher dimensions.

Let $Z_t =B_t +i \beta_t$ be a Brownian motion in $\mathbb{C}^n$ started at 0. This means that $B$ and $\beta$ are two independent Brownian motions in $\mathbb{R}^n$. We can then consider the one-form

$\alpha=\sum_{i=1}^n x_i dy_i -y_i dx_i$
and the generalized Levy area

$S_t =\int_{Z[0,t]} \alpha =\sum_{i=1}^n \int_0^t B^i_s d\beta^i_s -\beta^i_sdB_s^i$
The process

$X_t=(Z_t,S_t)$
is then a diffusion process in $\mathbb{C}^n \times \mathbb{R}$ with generator

$L=\frac{1}{2} \sum_{i=1}^n X_i^2+Y_i^2$
where

$X_i=\frac{\partial}{\partial x_i} -y_i \frac{\partial}{\partial z}$
$Y_i=\frac{\partial}{\partial y_i} +x_i \frac{\partial}{\partial z},$
$z$ being the last coordinate in $\mathbb{C}^n \times \mathbb{R}$. If we denote $Z= \frac{\partial}{\partial z}$, we have the following Lie brackets relations

$[X_i,X_j]=[Y_i,Y_j]=[X_i,Z]=[Y_j,Z]=0$
and

$[X_i,Y_j]=2 \delta_{ij} Z.$
In particular, the Hormander’s condition is satisfied. We also see that $X_i,Y_j,Z$ generate the Lie algebra of the $n$-dimensional Heisenberg group

$\mathbb{H}^{2n+1}=\{ (x,y,z) \in \mathbb{R}^n\times \mathbb{R}^n\times \mathbb{R} \}$
where the group law is given by

$(x_1,y_1,z_1) \star (x_2,y_2,z_2) =(x_1+x_2,y_2+y_2,z_1+z_2+\sum_{i=1}^n x^i_1y^i_2-x^i_2y^i_1).$
As before, we can then interpret $(X_t)_{t \ge 0}$ as a horizontal Brownian motion on $\mathbb{H}^{2n+1}$.

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

Consider for instance a general smooth one-form

$\alpha=\sum_{i=1}^n \alpha^i(x)dx_i$
on $\mathbb{R}^n$ and let $(B_t)_{t \ge 0}$ be a $n$-dimensional Brownian motion. We have

$\int_{B[0,t]} \alpha =\sum_{i=1}^n \int_0^t \alpha^i(B_s) \circ dB^i_s,$
where the stochastic integrals have to be understood in the Stratonovitch sense. 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 have

$[X_i,X_j]=\left( \frac{\partial \alpha^j}{\partial x_i}-\frac{\partial \alpha^i}{\partial x_j} \right) \frac{\partial}{\partial z}.$
Therefore, if the two-form $d\alpha$ is never 0, then the Hormander’s condition is satisfied.

We would like to call $(X_t)_{t \ge 0}$ the horizontal Brownian motion of some relevant geometric structure…

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