In the previous Lecture we proved that Brownian motion paths almost surely have a bounded -variation for every . In this lecture, we are going to prove that they even almost surely are -rough paths for . To prove this, we need to construct a geometric rough path over the Brownian motion, that is we need to lift the Brownian motion to the free nilpotent Lie group of step , . In this process, we will have to define the iterated integrals . This can be done by using the theory of stochastic integrals. Indeed, it is well known (and easy to prove !) that if
is a subdivision of the time interval whose mesh goes to , then the Riemann sums
converge in probability to a random variable denoted . We can then prove that the stochastic process admits a continuous version which is a martingale. With this integral of against itself in hands, we can now proceed to construct the canonical geometric rough path over .
Let and denote the space of skew-symmetric matrices. We can realize the group in the following way
where is the group law defined by
Here we use the following notation; if , then denotes the skew-symmetric matrix . Notice that the dilation writes
Remark: If is a continuous path with bounded variation then for we denote
where is the area swept out by the vector during the time interval . Then, it is easily checked that for ,
where is precisely the law of , i.e. for , ,
We now are in position to give the fundamental definition.
Definition: The process
is called the lift of the Brownian motion in the group .
Interestingly, it turns out that the lift of a Brownian motion is a Markov process. Indeed, consider the vector fields
defined on . It is easy to check that:
- For ,
- For ,
- The vector fields
are invariant with respect to the left action of on itself and form a basis of the Lie algebra of .
The process solves the Stratonovitch stochastic differential equation
and as such, is a diffusion process in whose generator is the subelliptic diffusion operator given by .
Finally, also observe that we have the following scaling property, for every $c> 0$,
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 be a metric space and be a continuous path. Let and . There exists a constant such that:
Theorem: The paths of are almost surely geometric -rough paths for . As a consequence, the Brownian motion paths almost surely are -rough paths for . Let .
Proof: We know that if , then . Therefore, we need to prove that for , the paths of almost surely have bounded -variation with respect to the Carnot-Caratheodory distance. From the scaling property of and of the Carnot-Caratheodory distance, we have in distribution
Moreover, from the equivalence of homogeneous norms, we have
It easily follows from that, that for every ,
Thus, from Fubini’s theorem we obtain
The Garsia-Rodemich-Rumsey inequality implies then
Therefore, the paths of almost surely have bounded -variation for