Global Analysis and Infinite-Dimensional Lie Groups.- Aspects of Stochastic Global Analysis.- A Lie Group Structure for Automorphisms of a Contact Weyl Manifold.- Riemannian Geometry.- Projective Structures of a Curve in a Conformal Space.- Deformations of Surfaces Preserving Conformal or Similarity Invariants.- Global Structures of Compact Conformally Flat Semi-Symmetric Spaces of Dimension 3 and of Non-Constant Curvature.- Differential Geometry of Analytic Surfaces with Singularities.- Symplectic Geometry and Poisson Geometry.- The Integration Problem for Complex Lie Algebroids.- Reduction, Induction and Ricci Flat Symplectic Connections.- Local Lie Algebra Determines Base Manifold.- Lie Algebroids Associated with Deformed Schouten Bracket of 2-Vector Fields.- Parabolic Geometries Associated with Differential Equations of Finite Type.- Quantizations and Noncommutative Geometry.- Toward Geometric Quantum Theory.- Resonance Gyrons and Quantum Geometry.- A Secondary Invariant of Foliated Spaces and Type III? von Neumann Algebras.- The Geometry of Space-Time and Its Deformations from a Physical Perspective.- Geometric Objects in an Approach to Quantum Geometry.
Aspects of Stochastic Global Analysis (P. 3)
K. D. Elworthy
Mathematics Institute, Warwick University, Coventry CV4 7AL, England
Dedicated to Hideki Omori
Summary.
This is a survey of some topics where global and stochastic analysis play a role. An introduction to analysis on Banach spaces with Gaussian measure leads to an analysis of the geometry of stochastic differential equations, stochastic flows, and their associated connections, with reference to some related topological vanishing theorems. Following that, there is a description of the construction of Sobolev calculi over path and loop spaces with diffusion measures, and also of a possible L2 de Rham and Hodge-Kodaira theory on path spaces. Diffeomorphism groups and diffusion measures on their path spaces are central to much of the discussion. Knowledge of stochastic analysis is not assumed.
Keywords:
Path space, diffeomorphism group, Hodge-Kodaira theory, in.nite dimensions, universal connection, stochastic differential equations, Malliavin calculus, Gaussian measures, differential forms, Weitzenbock formula, sub-Riemannian.
1 Introduction
Stochastic and global analysis come together in several distinct ways. One is from the fact that the basic objects of finite dimensional stochastic analysis naturally live on manifolds and often induce Riemannian or sub-Riemannian structures on those manifolds, so they have their own intrinsic geometry.
Another is that stochastic analysis is expected to be a major tool in infinite dimensional analysis because of the singularity of the operators which arise there, a fairly prevalent assumption has been that in this situation stochastic methods are more likely to be successful than direct attempts to extend PDE techniques to in.nite dimensional situations.
(Ironically that situation has been reversed in recent work on the stochastic 3D Navier–Stokes equation, [DPD03].) Stimulated particularly by the approach of Bismut to index theorems, [Bis84], and by other ideas from topology, representation theory, and theoretical physics, this has been extended to attempts to use stochastic analysis in the construction of ininite dimensional geometric structures, for example on loop spaces of Riemannian manifolds.
As examples see [AMT04], and [L´ea05]. In any case global analysis was firmly embedded in stochastic analysis with the advent of Malliavin calculus, a theory of Sobolev spaces and calculus on the space of continuous paths on Rn, as described briefly below, and especially its relationships with diffusion operators and processes on finite dimensional manifolds.
In this introductory selection of topics, both of these aspects of the intersection are touched on. After a brief introduction to analysis on spaces with Gaussian measure there is a discussion of the geometry of stochastic differential equations, stochastic flows, and their associated connections, with reference to some related topological vanishing theorems. Following that, there is a discussion of the construction of Sobolev calculi over path and loop groups with diffusion measures, and also of de Rham and Hodge-Kodaira theory on path spaces.
