De Rham complex over product manifolds: Dirichlet forms and stochastic dynamics (Q2712754)

Consider a compact smooth manifold \(M\), and the product \({\mathcal M}^\infty: =M^{\mathbb{Z}^d}\). The authors provide \({\mathcal M}^\infty\) with some subtangent bundle and a differentiable structure, and fix a probability measure \(\mu\) on \({\mathcal M}^\infty\), which satisfies an integration by parts formula, as do for example smooth Gibbs measures.NEWLINENEWLINENEWLINEThey then define the de Rham complex and the Bochner and de Rham Laplacians on \(({\mathcal M}^\infty,\mu)\), which satisfy a Weitzenböck formula.NEWLINENEWLINENEWLINEThen probabilistic representations of the corresponding semigroups are given, which allow to prove contraction properties.NEWLINENEWLINEFor the entire collection see [Zbl 0957.00063].