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

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