Measures and Differential Equations in Infinite-Dimensional Space

Inhaltsverzeichnis

• I. Measures and quasimeasures. Integration.
• 1. Realvalued measures on algebras of sets.
• 2. Cylinder sets and cylindrical functions.
• 3. Quasimeasures. Integration.
• 4. Supplement: Some notions related to the topology of linear spaces.
• 5. Chapter I: Supplementary remarks and historical comments.
• II. Gaussian measures in Hilbert space.
• 1. Gaussian measures in finite-dimensional spaces.
• 2. Gaussian measures in Hilbert space.
• 3. Measurable linear functionals and operators.
• 4. Absolute continuity of Gaussian measures.
• 5. Fourier-Wiener transformation.
• 6. Complexvalued Gaussian quasimeasures.
• 7. Chapter II: Supplementary re marks and historical comments.
• III. Measures in linear topological spaces.
• 1. ?-additivity conditions for nonnegative cylindrical measures in the space X' dual to a locally convex space X.
• 2. Sequences of Radon measures.
• 3. Chapter III: Supplementary remarks and historical comments.
• IV. Differentiable measures and distributions.
• 1. Differentiable functions, differentiable expressions.
• 2. Differentiable measures.
• 3. Distributions and generalized functions.
• 4. Positive definiteness. Quasi-invariant distributions and bidistributions.
• 5. Chapter IV: Supplementary remarks and historical comments.
• V. Evolution differential equations.
• 1. Weak solutions of evolution equations.
• 2. Second order equations with variable coefficient.
• 3. Chapter V: Supplementary remarks and historical comments.
• VI. Integration in path space.
• 1. Markov quasimeasures.
• 2. Evolution families of operators.
• 3. Linear evolution families and functional integrals.
• 4. Nonlinear evolution families, and integrals in branching path space.
• 5. Chapter VI: Supplementary remarks and historical comments.
• VII. Probabilistic representations of solutions of parabolicequations and systems.
• 1. Wiener process. Stochastic integrals.
• 2. Stochastic differential equations.
• 3. Operator multiplicative functionals and the evolution families determined by them.
• 4. The Cauchy problem for second order parabolic systems.
• 5. Chapter VII: Supplementary remarks and historical comments.
• VIII. Smooth measures.
• 1. Admissible operators.
• 2. Admissibility of differential operators.
• 3. Absolute continuity of smooth measures.
• 4. Maps of spaces and admissible operators.
• 5. Biorthogonal systems in L2(X, ?).
• 6. Chapter VIII: Supplementary remarks and historical comments.
• Supplement to chapters IV-V.
• 1. Essentially infinite-dimensional elliptic operators.
• 2. Properties of essentially infinite-dimensional elliptic operators, and solutions of the corresponding Cauchy problem.
• 3. Existence of solutions of the Cauchy problem.
• 4. Supplementary remarks and historical comments.
• Supplement to chapter VII.
• 1. Supplementary remarks and historical comments.