It is now time to give a fundamental example of rough path: The Brownian motion. As we are going to see, a Brownian motion is a -rough path for any .
We first remind the following basic definition.
Definition: Let be a probability space. A continuous -dimensional process is called a standard Brownian motion if it is a Gaussian process with mean function
and covariance function
For a Brownian motion , the following properties are easy to check:
- For any , the process is a standard Brownian motion;
- For any , the random variable is independent of the -algebra .
- For every , the process has the same law as the process .
An easy computation shows that for and :
Therefore, as a consequence of the Kolmogorov continuity theorem, for any and , there exists a finite random variable such that for ,
We deduce in particular that for any , we have almost surely
We now prove that for , we have almost surely
In the sequel, if
is a subdivision of the time interval , we denote by
the mesh of this subdivision.
Proposition: Let be a standard Brownian motion. Let . For every sequence of subdivisions such that
the following convergence takes place in (and thus in probability),
As a consequence, if , for every , almost surely,
Proof: We prove the result in dimension 1 and let the reader adapt it to the multidimensional setting.
Let us denote
Thanks to the stationarity and the independence of Brownian increments, we have:
Let us now prove that, as a consequence of this convergence, the paths of the process almost surely have an infinite -variation on the time interval if . Reasoning by absurd, let us assume that . From the above result, since the convergence in probability implies the existence of an almost surely convergent subsequence, we can find a sequence of subdivisions whose mesh tends to and such that almost surely,
We get then
which is clearly absurd
Therefore only the case is let open. It is actually possible to prove that:
Proposition: For every , we have almost surely
Proof: See the book by Friz-Victoir page 381