Stochastic Analysis in Infinitely Dimensional Spaces and quasi-invariance of Wiener measures

We will use the quasi-invariance problem of Wiener measures as an example to explain the special difficulties of analysis in infinite dimensional spaces that are not present in finite dimensional spaces. This problem becomes particularly interesting if we consider an infinite-dimensional space that is not flat. Typical examples are path and loop spaces over a Riemannian manifold (e.g., a sphere). In particular, we will show that the presence of curvature of the base space will result in an orthogonal rotation (in a flat space) for which the Wiener measure is (fortunately) invariant.