Let us consider a differential equation that writes
where the ‘s are vector fields on and where the driving signal is a continuous bounded variation path. If the vector fields are Lipschitz continuous then, for any fixed initial condition, there is a unique solution to the equation. We can see this solution as a function of the driving signal . It is an important question to understand for which topology, this function is continuous.
A simple example shows that the topology of uniform convergence is not the correct one here. Indeed, let us consider the differential equation
A straightforward computation shows that . When , converges uniformly to 0 whereas, of course, does not converge to 0. In this framework , a correct topology is given by the topology of convergence in 1-variation on compact sets. To fix the ideas, let us work on the interval . The distance in 1-variation between two continuous bounded variation paths is given by
where the supremum is taken over all the subdivisions
It is then a fact that is going to be proved in this class that if the ‘s are bounded and if is a sequence of bounded variation paths that converges in 1-variation to a continuous path with bounded variation, then the solutions of the differential equations
converge in 1-variation to the solution of
This type of continuity result suggests to use a topology in -variation, , to try to extend the map to a larger class of driving signals . More precisely, for , let us denote by the closure of the set of continuous with bounded variation paths with respect to the distance in -variation which is given by
We will then prove the following result:
Proposition: Let . If is a sequence of bounded variation paths that converges in -variation to a path , then the solutions of the differential equations
converge in -variation to some . Moreover is the solution of the differential equation
where the integrals are understood in the sense of Young’s integration.
The value is really a treshold: The result is simply false for . The main idea of the rough paths theory is to introduce a much stronger topology than the convergence in -variation. This topology, that we now explain, is related to the continuity of lifts of paths in free nilpotent Lie groups.
Let be the free -step nilpotent Lie group with generators . If is continuous with bounded variation, the solution of the equation
is called the lift of in . For , let us
denote the closure of the set of lifted paths with respect to the distance in -variation which is given by
where denotes the Carnot-Caratheodory distance on the group . This is a distance that will be explained in details later. Its main property is that it is homogeneous with respect to the natural dilation of .
Consider now the map which associates to a continuous with bounded variation path
the continuous path with bounded variation that solves the ordinary differential equation
It is clear that there exists a unique map from the set of continuous with bounded variation lifted paths onto the set of continuous with bounded variation lifted paths which makes the following diagram commutative
The fundamental theorem of Lyons is the following:
Theorem: If , then in the topology of -variation, there exists a continuous extension of from into .
In particular, we can now give a sense to differential equations driven by some continuous paths with finite -variation, for any . Indeed, let which is continuous with a fnite -variation and assume that there exists whose projection onto is . The projection onto of is then understood as being a solution of
An important example of application is given by the case where the driving signal is a Brownian motion . Brownian motion has a -finite variation for any and, as we will see, admits a canonical lift in . As a conclusion, we can consider in the rough paths sense, solutions to the equation
It turns out that this notion of solution is exactly equivalent to solutions that are obtained by using the Stratonovitch integration theory. Therefore, the theory of stochastic differential equations appears as a very special case of the rough paths theory !