Stochastic Equations and Differential Geometry

Inhaltsverzeichnis

• 1. Functions and Measures in Linear Spaces.
• 1. Spaces, Mappings, Differential Operations.
• 2. Measures and Integrals.
• 3. Measure Differentiation.
• 2. Functions and Measures on Smooth Manifolds.
• 1. Smooth Manifolds and Vector Bundles.
• 2. Bundle Section, Connections, Differential Operations.
• 3. Hilbert and Hilbert-Schmidt Bundles.
• 4. Measures on Smooth Manifolds.
• 3. Stochastic Equations in Banach Spaces.
• 1. Basic Notions.
• 2. Stochastic Integrals for Vector and Operator Functions.
• 3. Stochastic Equations.
• 4. Multiplicative Functionals of Stochastic Processes.
• 5. Stochastic Flow.
• 4. Stochastic Equations on Smooth Manifolds.
• 1. Stochastic Differentials.
• 2. Stochastic Differential Equations on Manifolds.
• 3. Stochastic Equations in Vector Bundles.
• 5. Kolmogorov Equations.
• 1. Backward Kolmogorov Equations.
• 2. Quasilinear Parabolic Equations.
• 3. Forward Kolmogorov Equations.
• 6. Diffusion Processes on Lie Groups and Principal Fibre Bundles.
• 1. Lie Groups and Smooth Bundles.
• 2. Invariant Stochastic Equations.
• 3. Stochastic Equations on Manifolds and their Solution Distribution Properties.
• Historical Comments.
• References.