The St Flour Lectures on Rough Paths took place between Friday 9th July and Saturday the 24th July inclusive. There were ten lectures of 90 Minutes.
Here are my very rough hand written notes distributed to the participants before each lecture. They have many detailed misprints and errors - most of which were corrected at the time - but obviously not in these (which were the contemporaneous versions) notes. None the less, the plan is to put these notes into a more respectable and presentable form before they appear as part of the St Flour series and so, if anyone reads them and notices errors it would be very helpful to send them to the author. A convenient way to communicate could be to simply write your comments on the relevant page and fax it to me. (+44 1865 273583 for attention terry lyons). Many thanks!!
A mathematical point: if you do work through these notes despite the misprints, then i hope they will provide a brief and reasonably motivated account of rough paths that would equip one to study the published work in the area or my book with Zhongmin Qian. But please also be aware that the notation has not stabilised largely because of two scientific reasons.
The first is the role of geometric rough paths - should they be what I call \Omega_{0,p} or what I call \Omega G_p and what notation should one use for the choice. These preliminary notes do not help in the matter although they do make it clear that the ability to approximate in the p-metric is rarely the crucial issue - rather it is the ability to approximate in a weaker metric to establish certain identities - and that it is these identities that ensure the convergence and continuity theorems in rough path theory; so \Omega G_p, as defined here, is certainly a useful space. Unfortunately I blurred the distinctions between these concepts ten years ago, which has lead to different notations.
A second issue is what is the correct definition of closeness for rough paths. There is the definition used here in terms of iterated integral being close, and there is also the definition in terms of paths of bounded p-variation in a group. In finite dimensions they are equivalent. However, I am not sure the concepts do not diverge in infinite dimensions, in which case in which case I believe that the notion of closeness in terms of iterated integrals is the correct one even if it is less elegant, as it does not require any implicit calculation of inverses and seems analytically more robust; in any case the theorems work for it without reference to dimension. On the other hand I do believe that, in any case, the equivalence between the two notions in finite dimensions is a helpful one.
I will try to ensure both issues are addressed in a way that maximises consistency when the notes are finally published.
Finally, these notes were real notes for a real lecture course intended to get people going and were prepared to a large part without access to a library; in particular, many people have contributed to the development of results in these notes - and the reader should not assume that material written here is work of the author. Many credits were given aurally.
Finally - THANKS to the tremendous audience who participated and followed some very intensive and advanced lecturees and worked hard between them to understand. The quality and penetration of their comments made the experience a temendous one.
You should be able to open the notes as web pages from these links:
st flour notes/Lecture One.mht
st flour notes/Lecture Two.mht
st flour notes/Lecture Three.mht
st flour notes/Lecture Four.mht
st flour notes/Lecture Five.mht
st flour notes/Lecture Six.mht
st flour notes/Lecture Seven.mht
st flour notes/Lecture Eight.mht
st flour notes/Lecture Nine and ten.mht
st flour notes/Lecture Ten bis.mht
Or as massive PDF files from these links:
Lecture note images/Lecture one .pdf
Lecture note images/Lecture two .pdf
Lecture note images/Lecture three .pdf
Lecture note images/Lecture four.pdf
Lecture note images/Lecture five.pdf
Lecture note images/Lecture six.pdf
Lecture note images/Lecture seven.pdf
Lecture note images/Lecture eight.pdf
Lecture note images/Lecture nine and ten .pdf
Lecture note images/lecture10bis.pdf
The broad outline of the course were provided with background reading before
the meeting
course outline
and a bibliography for background reading:
:Rough Paths Bibliography