Lecture 21. The Brownian motion as a rough path (2)

In the previous Lecture we proved that Brownian motion paths almost surely have a bounded $p$ -variation for every $p > 2$ . In this lecture, we are going to prove that they even almost surely are $p$ -rough paths for $2 < p < 3$ . To prove this, we need to construct a geometric $p$ rough path over the Brownian motion, that is we need to lift the Brownian motion to the free nilpotent Lie group of step $2$ , $\mathbb{G}_{2} (\mathbb{R}^d)$ . In this process, we will have to define the iterated integrals $\int dB^{\otimes 2}=\int B \otimes dB$ . This can be done by using the theory of stochastic integrals. Indeed, it is well known (and easy to prove !) that if
$\Delta_n [0,t]=\left\{ 0=t^n_0 \le t^n_1 \le ...\le t^n_n=t \right\}$
is a subdivision of the time interval $[0,t]$ whose mesh goes to $0$ , then the Riemann sums
$\sum_{k=0}^{n-1} B_{t_k^n} \otimes (B_{t_{k+1}^n}-B_{t_k^n})$
converge in probability to a random variable denoted $\int_0^t B_s \otimes dB_s$ . We can then prove that the stochastic process $\int_0^t B_s \otimes dB_s$ admits a continuous version which is a martingale. With this integral of $B$ against itself in hands, we can now proceed to construct the canonical geometric rough path over $B$ .

Let $d \geq 2$ and denote $\mathcal{AS}_d$ the space of $d \times d$ skew-symmetric matrices. We can realize the group $\mathbb{G}_{2} (\mathbb{R}^d )$ in the following way
$\mathbb{G}_{2} (\mathbb{R}^d ) = ( \mathbb{R}^d \times \mathcal{AS}_d ,\circledast)$
where $\circledast$ is the group law defined by
$( \alpha_1 , \omega_1 ) \circledast ( \alpha_2 , \omega_2 )= ( \alpha_1 + \alpha_2 , \omega_1 + \omega_2 + \frac{1}{2} \alpha_1 \wedge \alpha_2 ).$
Here we use the following notation; if $\alpha_1, \alpha_2 \in \mathbb{R}^d$ , then $\alpha_1 \wedge \alpha_2$ denotes the skew-symmetric matrix $\left( \alpha_1^i \alpha_2^j - \alpha_1^j \alpha_2^i \right)_{i,j}$ . Notice that the dilation writes
$\label{scaling 2 step} c \cdot ( \alpha , \omega ) = ( c \alpha , c^2 \omega ).$

Remark: If $x:[0,+\infty) \rightarrow \mathbb{R}^2$ is a continuous path with bounded variation then for $0 < t_1 < t_2$ we denote
$\Delta_{[t_1,t_2]}x=\left( x^1_{t_2}-x^1_{t_1},x^2_{t_2}-x^2_{t_1},S_{[t_1,t_2]}x \right),$
where $S_{[t_1,t_2]}x$ is the area swept out by the vector $\overrightarrow{x_{t_1}x_t}$ during the time interval $[t_1,t_2]$ . Then, it is easily checked that for $0 < t_1 < t_2 < t_3$ ,
$\Delta_{[t_1,t_3]}x=\Delta_{[t_1,t_2]}x \circledast \Delta_{[t_2,t_3]}x,$
where $\circledast$ is precisely the law of $\mathbb{G}_{2} (\mathbb{R}^2 )$ , i.e. for $(x_1,y_1,z_1)$ , $(x_2,y_2,z_2) \in \mathbb{R}^3$ ,
$(x_1,y_1,z_1) \circledast (x_2,y_2,z_2)=\left( x_1+x_2,y_1+y_2,z_1+z_2+\frac{1}{2} \left(x_1 y_2 - x_2 y_1 \right) \right).$

We now are in position to give the fundamental definition.
Definition: The process
$\mathbf{B}_{t}=\left( B_t , \frac{1}{2} \left( \int_0^t B^i_s dB^j_s-B^j_s dB^i_s \right)_{1 \leq i,j \leq d} \right), \text{ }t \geq 0.$
is called the lift of the Brownian motion $(B_{t})_{ t \geq 0}$ in the group $\mathbb{G}_{2}(\mathbb{R}^d )$ .

Interestingly, it turns out that the lift of a Brownian motion is a Markov process. Indeed, consider the vector fields
$D_i (x)=\frac{\partial}{\partial x^i}+ \frac{1}{2} \sum_{j < i} x^j \frac{\partial}{\partial x^{j,i}}- \frac{1}{2} \sum_{j > i} x^j \frac{\partial}{\partial x^{i,j}}, \text{ }1 \leq i \leq d,$
defined on $\mathbb{R}^d \times \mathcal{AS}_d$ . It is easy to check that:

For $x \in \mathbb{R}^d \times \mathcal{AS}_d$ ,
$[ D_i , D_j ](x)= \frac{\partial}{\partial x^{i,j}}, \text{ } 1 \leq i < j \leq d;$
For $x \in \mathbb{R}^d \times \mathcal{AS}_d$ ,
$[[ D_i , D_j],D_k ](x)= 0, \text{ }1 \leq i ,j,k \leq d;$
The vector fields $\left( D_i , [ D_j , D_k ] \right)_{1 \leq i \leq d, 1 \leq j < k \leq d}$
are invariant with respect to the left action of $\mathbb{G}_{2} (\mathbb{R}^d )$ on itself and form a basis of the Lie algebra $\mathfrak{g}_{2} (\mathbb{R}^d)$ of $\mathbb{G}_{2} (\mathbb{R}^d )$ .

The process $(\mathbf{B}_{t})_{ t \geq 0}$ solves the Stratonovitch stochastic differential equation
$d\mathbf{B}_t=\sum_{i=1}^d D_i (\mathbf{B}_t) \circ dB^i_s.$
and as such, is a diffusion process in $\mathbb{R}^d \times \mathcal{AS}_d$ whose generator is the subelliptic diffusion operator given by $\sum_{i=1}^d D_i^2$ .

Finally, also observe that we have the following scaling property, for every $c> 0$,
$\left( \mathbf{B}_{ct} \right)_{t \geq 0} =^{\text{law}} \left(\sqrt{c} \cdot \mathbf{B}_{t} \right)_{t \geq 0}.$

Before we turn to the fundamental result of this Lecture, we need the following result which is known as the Garsia-Rodemich-Rumsey inequality (see the proof page 573 in the book by Friz-Victoir):

Lemma: Let $(X,d)$ be a metric space and $x:[0,T] \to E$ be a continuous path. Let $q > 1$ and $\alpha \in (1/q,1)$ . There exists a constant $C=C(\alpha,q)$ such that:
$d(x(s),x(t))^q\le C |t-s|^{\alpha q -1} \int_{[s,t]^2} \frac{ d(x(u),x(v))^q}{|u-v|^{1+\alpha q}} du dv.$

Theorem: The paths of $(\mathbf{B}_{t})_{ t \geq 0}$ are almost surely geometric $p$ -rough paths for $2 < p < 3$ . As a consequence, the Brownian motion paths almost surely are $p$ -rough paths for $2 < p < 3$ . Let $q > 1$ .

Proof: We know that if $q < p$ , then $C_0^{q-var} ([0,T], \mathbb{G}_{[p]} (\mathbb{R}^d)) \subset \mathbf{\Omega G}^p([0,T],\mathbb{R}^d)$ . Therefore, we need to prove that for $2 < p < 3$ , the paths of $(\mathbf{B}_{t})_{ t \geq 0}$ almost surely have bounded $p$ -variation with respect to the Carnot-Caratheodory distance. From the scaling property of $(\mathbf{B}_{t})_{ t \geq 0}$ and of the Carnot-Caratheodory distance, we have in distribution
$d( \mathbf{B}_{s},\mathbf{B}_{t})=^d \sqrt{ t-s} d (0, \mathbf{B}_1).$
Moreover, from the equivalence of homogeneous norms, we have
$d (0, \mathbf{B}_1) \simeq \| B_1\| + \left\| \int_0^1 B \otimes dB \right\|^{1/2}.$
It easily follows from that, that for every $q > 1$ ,
$\mathbb{E} \left( \frac{ d( \mathbf{B}_{s},\mathbf{B}_{t})^q }{ (t-s)^{q/2} } \right)=\mathbb{E} \left( d (0, \mathbf{B}_1)^q\right) < +\infty.$
Thus, from Fubini’s theorem we obtain
$\mathbb{E} \left( \int_{[0,T]^2} \frac{ d(\mathbf{B}_{u},\mathbf{B}_{v})^q}{|u-v|^{q/2}} du dv \right)< +\infty.$
The Garsia-Rodemich-Rumsey inequality implies then
$d( \mathbf{B}_{s},\mathbf{B}_{t})^q\le C |t-s|^{q/2 -1} \int_{[0,T]^2} \frac{ d(\mathbf{B}_{u},\mathbf{B}_{v})^q}{|u-v|^{q/2}} du dv.$
Therefore, the paths of $(\mathbf{B}_{t})_{ t \geq 0}$ almost surely have bounded $p$ -variation for $p > 2$ $\square$