[For the entire collection see Zbl 0733.00017.]
The paper is a profound investigation on the stochastic \(L^ p\)-Chen forms on loop spaces.
Let \(M\) be a Riemannian manifold. The authors study differential forms on the following infinite dimensional manifolds: (1) the path space \(PM\) consisting of paths \(w:[0,1]\to M\), (2) the loop space \(LM\) consisting of paths \(w\) such that \(w(0)=w(1)\), (3) the based loop space \(L_ xM\) consisting of loops \(w\) such that \(w(0)=w(1)=x\) where \(x\) is a chosen base point in \(M\). The theory of stochastic Chen forms is described in paragraph 1. In paragraph 2 and 3 the authors introduce natural \(L^ p\)- norms on the space of forms on \(LM\) and study the completions of the space of Chen forms in these \(L^ p\)-norms. These completed spaces are the spaces of \(L^ p\)-Chen forms. Paragraph 4 and 5 contain estimates on which the work is based.