Author Archives: Fabrice Baudoin

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 … Continue reading →

In this Lecture, the geometric concepts introduced in the previous lectures are now used to revisit the notion of -rough path that was introduced before. We will see that using Carnot groups gives a perfect description of the space of … Continue reading →

In this Lecture, we go one step further to understand -rough paths from paths in Carnot groups. The connection is made through the study of paths with bounded p-variation in Carnot groups. Definition: A continuous path is said to have … Continue reading →

In this Lecture we introduce a canonical distance on a Carnot group. This distance is naturally associated to the sub-Riemannian structure which is carried by a Carnot group. It plays a fundamental role in the rough paths topology. Let be … Continue reading →

We introduce here the notion of Carnot group, which is the correct structure to understand the algebra of the iterated integrals of a path up to a given order. It is worth mentioning that these groups play a fundamental role … Continue reading →

In the previous lecture, we proved the Chen’s expansion formula which establishes the fact that the signature of a path is the exponential of a Lie series. This expansion is of course formal but analytically makes sense in a number … Continue reading →

The next few lectures will be devoted to the construction of the so-called geometric rough paths. These paths are the lifts of the -rough paths in the free nilpotent Lie group of order . The construction which is of algebraic … Continue reading →

In this lecture we define solutions of linear differential equations driven by -rough paths, and present the Lyons’ continuity theorem in this setting. Let be a -rough path with truncated signature and let be an approximating sequence such that Let … Continue reading →

In this lecture, it is now time to harvest the fruits of the two previous lectures. This will allow us to finally define the notion of -rough path and to construct the signature of such path. A first result which … Continue reading →

Let . Since is a control, the estimate easily implies that for , We stress that it does not imply a bound on the 1-variation of the path . What we can get for this path, are bounds in -variation: … Continue reading →