Lecture 2. Horizontal Brownian motion on the Heisenberg group

Posted on July 12, 2016 by Fabrice Baudoin

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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