Stochastic Analysis on Product Manifolds: Dirichlet Operators on Differential Forms

Abstract

We define a de Rham complex over a product manifold (infinite product of compact manifolds), and Dirichlet operators on differential forms, associated with differentiable measures (in particular, Gibbs measures), which generalize the notions of Bochner and de Rham Laplacians. We give probabilistic representations for corresponding semigroups and study properties of the corresponding stochastic dynamics.

Supersymmetric Dirichlet Operators, Spectral Gaps, and Correlations

“…It makes use of the supersymmetric structure of L β which implies that L (0) β , restricted to the orthogonal complement in L 2 (μ β ) of its kernel, is unitarily equivalent to L (1) β , restricted to the range ofd β L 2 (μ β ) , provided the latter is closed [32]. In particular, apart from zero the spectrum of L (0) β is contained in the spectrum of L (1) β . The second main ingredient in the proof is the low-temperature localization mentioned above.…”

“…It is expressed by a relative form bound involving L β , which is another result of Section 7. In particular, for large β, L (1) β turns out to be a small form perturbation of L (0)…”

“…Knowing the spectrum of L (0) β up to some number E we thus get information about the spectrum of L (1) β up to E + sup σ(H (0)). All spectral gaps of L (1) β determined in this way must also be spectral gaps of L (0) β and we may proceed on inductively.…”

“…Knowing the spectrum of L (0) β up to some number E we thus get information about the spectrum of L (1) β up to E + sup σ(H (0)). All spectral gaps of L (1) β determined in this way must also be spectral gaps of L (0) β and we may proceed on inductively. This type of reasoning goes back to [49], where higher order Grushin problems are considered, and has been extended to higher form degrees in [40].…”

$L^2$-Betti Numbers of Infinite Configuration Spaces

“…Hypersurfaces in the Wiener space were considered in [35]. The de Rham complex on infinite product manifolds with Gibbs measures (which appear in connection with problems of classical statistical mechanics) was constructed in [1], [2] (see also [19] for the case of the infinite-dimensional torus). We should also mention the papers [49], [15], [16], [17], [7], where the case of a flat Hilbert state space is considered (the L 2 -cohomological structure turns out to be nontrivial even in this case due to the existence of interesting measures on such a space).…”