Stochastic Analysis on Product Manifolds: Dirichlet Operators on Differential Forms
✍ Scribed by Sergio Albeverio; Alexei Daletskii; Yuri Kondratiev
- Publisher
- Elsevier Science
- Year
- 2000
- Tongue
- English
- Weight
- 263 KB
- Volume
- 176
- Category
- Article
- ISSN
- 0022-1236
No coin nor oath required. For personal study only.
✦ Synopsis
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. 2000 Academic Press Contents. 1. Introduction. 2. Setting. 3. Differential forms and the de Rham complex. 4. Dirichlet operators on differentiable forms. 4.1. Dirichlet forms and stochastic dynamics on functions. 4.2. Dirichlet forms and Dirichlet operators in spaces of differential forms, and associated stochastic dynamics. 5. Probabilistic representations of semigroups. 5.1. Stochastic differential equations on product manifolds. 5.2. Parallel translation and diffusions on tensor bundles. 5.3. Probabilistic representations of semigroups. 6. Stochastic dynamics for lattice models associated with Gibbs measures on product manifolds. 6.1. Gibbs measures on product manifolds. 6.2. Stochastic dynamics on functions. 6.3. Stochastic dynamics in the de Rham complex. 7. Appendix. 7.1. Differentiable structures on product manifolds. 7.2. Existence and uniqueness of solutions for infinite systems of SDE.