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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