Author Archives: Fabrice Baudoin

In the previous lecture we introduced the signature of a bounded variation path as the formal series If now , the iterated integrals can only be defined as Young integrals when . In this lecture, we are going to derive … Continue reading →

In this lecture we introduce the central notion of the signature of a path which is a convenient way to encode all the algebraic information on the path which is relevant to study differential equations driven by . The motivation … Continue reading →

In the previous lecture we defined the Young’s integral when and with . The integral path has then a bounded -variation. Now, if is a Lipschitz map, then the integral, is only defined when , that is for . With … Continue reading →

In this lecture we define the Young‘s integral when and with . The cornerstone is the following Young-Loeve estimate. Theorem: Let and . Consider now with . The following estimate holds: for , Proof: For , let us define We … Continue reading →

Our next goal in this course is to define an integral that can be used to integrate rougher paths than bounded variation. As we are going to see, Young’s integration theory allows to define as soon as has finite -variation … Continue reading →

Let and let be a Lipschitz continuous map. In order to analyse the solution of the differential equation, and make the geometry enter into the scene, it is convenient to see as a collection of vector fields , where the … Continue reading →

In this lecture we establish the basic existence and uniqueness results concerning differential equations driven by bounded variation paths and prove the continuity in the 1-variation topology of the solution of an equation with respect to the driving signal. Theorem: … Continue reading →

Let be a piecewise continuous path and . It is well-known that we can integrate against by using the Riemann–Stieltjes integral which is a natural extension of the Riemann integral. The idea is to use the Riemann sums where . … Continue reading →

The first few lectures are essentially reminders of undergraduate real analysis materials. We will cover some aspects of the theory of differential equations driven by continuous paths with bounded variation. The point is to fix some notations that will be … Continue reading →

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 … Continue reading →