Special Itô maps and an \(L^2\) Hodge theory (Q2702439)

The authors investigate \(L^2\) differential forms on (continuous) path spaces over a smooth compact Riemannian manifold. The general aim is to construct differential analysis, associated with Brownian motion measure and admitting an analogue of Hodge theory. The new point in the paper is the use of a special Hilbert space \({\mathcal H}^2_\sigma\) continuously embedded into \(\Lambda^2\) of the tangent space to the path space at \(\sigma\). This allows the authors to succeed in constructing a well-posed Hodge theory for 1-forms. The technique involves Ito maps of gradient systems, and some facts of independent interest for these are also obtained.NEWLINENEWLINEFor the entire collection see [Zbl 0953.00048].