Zhongmin Qian

This chapter studies spaces of paths as geometric objects in their own right, and particularly looks at the possibility of constructing interesting and mathematically meaningful vector fields on these infinite dimensional spaces. A natural example of a vector field on path space can be obtained by solving a differential equation along the path. In this way, one can obtain a new direction at every point along the path and one might try to deform the path in these directions. Formally, such vector fields on path space are called Driver vector fields. Because the Itô functional is so discontinuous, these vector fields have very bad properties when one takes a classical perspective on path space. However, the Driver vector fields are far more regular when regarded as vector fields on rough path space. This chapter proves that one can actually solve the ordinary differential equations that they define and produce flows on path space corresponding to Driver vector fields. By constructing these flows, one is implicitly constructing a solution to a rough non linear equation. One should not be surprised that, in general, the solutions blow up after a short time interval.

This chapter establishes an integration theory for rough paths. The key result is the existence of the integral of a Lipschitz one-form against a rough path. In particular, this gives a change of variable formula, and allows one to see that the notion of a rough path is a geometric concept, stable under smooth change of variable. A key technical point is the use of the notion of Lipschitz, to be found in Stein's book ‘Singular integrals’, which allows for functions to be Lipschitz of any degree. The basic integration result requires the one-form to be Lipschitz of degree > p-1, in order to integrate to against all p-rough paths.

This chapter proves that the dyadic polygonal approximations to a Brownian paths, with probability 1, a Cauchy sequence of rough paths, and in this way, the chapter proves a result first established in its simplest form by E. Sipilainen, that Brownian motion, with Lévy area, can be regarded as a rough path. This basic result can be extended to Brownian motion in Banach spaces and to fractional Brownian motions with Hurst index < 4. Since the notion of rough paths depends on computing iterated integrals, and these live in tensor product spaces, any discussion of rough paths in infinite dimensional contexts requires knowledge of different tensor norms. Brownian motion in a Banach space is always a rough path in the injective tensor norm, and normally a rough path in the projective tensor norm. Indeed, it is an open problem to construct an example where the Lévy area did not exist in the projective norm.

This chapter defines rough paths. If the path has good smoothness properties, then its chords (the Abelian version of the description) provide an adequate description and can be used to predict its effects on a controlled system. A rough path uses a nilpotent group element, computed using Chen iterated integrals as an extended description. The chapter introduces the notion of a control and proves several basic results for paths whose nilpotent descriptors are appropriately controlled, and gives the formal definition of a rough path. Key theorems are proved. In particular, the extension theorem allowing one to compute all iterated integrals of a rough path, and the notion of an almost multiplicative functional, which is important for the development of an integration theory, are both introduced.

The theory of controlled systems is worked out in detail for the case where the driving stimulus or noise is a Lipschitz function. The Itô functional, which is the map from stimulus to response, is shown to be continuous as one varies the stimulus in the Lipshitz metric. Counter examples are given to show that the Itô functional is discontinuous in p-variation or Hölder norms, where p > 2, or the Hölder norm refers to a fractional power < ½.

The concept of a differential equation controlled by a rough path can be motivated by quite simple examples. One such example is a linear system driven by two-dimensional noise. This example is developed and an explicit answer is given to the question. Exactly what information should I extract from the driving stimulus or noise in order to accurately predict the response? The notions of a controlled system of Chen's iterated integral are introduced. The main notion in this book is the concept of a rough path. Almost all paths that one encounters in everyday life are only described approximately. Newton observed that a smooth path is actually quite well approximated by its chords. If one wants to describe a path γ over a short time interval from s to t, then it is enough to evaluate γ at these two times and consider the approximation that comes from replacing γ by the straight line with the same increment. This approach is not adequate if the control or path γ is oscillatory on the scale witnessed by the times s and t. If the path γ represented a text, then the chord is simply a word count. It turns out that a better description, one which takes into account the order of the events represented by γ, can be achieved by a description of γ that involves its first few Chen iterated integrals.

Several new Harnack estimates for positive solutions of the heat equation on a complete Riemannian manifold with Ricci curvature bounded below by a positive or a negative constant are established. These estimates are sharp both for small time for large time and for large distance and lead to new estimates for the heat kernel of a manifold with Ricci curvature bounded below.