Stochastic Geometry Primer

A Primer on Riemannian Geometry and Stochastic Analysis on Path Spaces

(Lecture notes of the ``mini course'' that the author gave at the ETH (Z\"urich) and the University of Z\"urich in February of 1995.)

I am sorry to say this file does not contain the pictures which were hand drawn in the hard copy versions. There are also a number of typographical errors in these notes. Hopefully sometime in the future I will have time to fix these problems. So for now use at your own risk.

\begin{abstract} These notes represent an expanded version of the ``mini course'' that the author gave at the ETH (Z\"urich) and the University of Z\"urich in February of 1995. The purpose of these notes is to provide some basic background to Riemannian geometry, stochastic calculus on manifolds, and infinite dimensional analysis on path spaces. No differential geometry is assumed. However, it is assumed that the reader is comfortable with stochastic calculus and differential equations on Euclidean spaces.

Let me summarize the contents of the talks at the ETH and Z\"urich.
\begin{enumerate} \item The first talk was on an extension of the Cameron Martin quasi-invariance theorem to manifolds. This lecture is not contained in these notes. The interested reader may consult Driver \cite{D5,D6} for the original papers. For more expository papers on this topic see \cite{D7,D9}. (These papers are complimentary to these notes.) The reader should also consult Hsu \cite{Hsu1}, Norris \cite{No3}, and Enchev and Stroock \cite{ES1,ES2} for the state of the art in this topic.

\item The second lecture encompassed sections 1-2.3 of these notes. This is an introduction to embedded submanifolds and the Riemannian geometry on them which is induced from the ambient space.

\item The third lecture covered sections 2.4-2.7. The topics were parallel translation, the development map, and the differential of the development map. This was all done for smooth paths.

\item\ The fourth lecture covered parts of sections 3 and 4. Here we touched on stochastic development map and its differential. Integration by parts formula for the path space and some spectral properties of an ``Ornstein-Uhlenbeck'' like operator on the path space.

\end{enumerate}

The manuscript is available as a PDF file (600K). Click here to retrieve.

11/26/2018 10:37 AM