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# Measures and Differential Equations in Infinite-Dimensional Space

## byYu.L. Dalecky, S.V. Fomin

### Paperback | October 26, 2012

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lEt moi, .... si j'avait Sll comment en revenir, One service mathematics has rendered the human race. It has put common sense back je n'y serais point aile:' where it belongs, on the topmost shelf next Jules Verne to the dusty canister labelled 'discarded 0- sense'. The series is divergent; therefore we may be Eric T. Bell able to do something with it. o. Heaviside Mathematics is a tool for thought. A highly necessary tool in a world where both feedback and non linearities abound. Similarly, all kinds of parts of mathematics serve as tools for other parts and for other sciences. Applying a simple rewriting rule to the quote on the right above one finds such statements as: 'One service topology has rendered mathematical physics .. .'; 'One service logic has rendered com puter science .. .'; 'One service category theory has rendered mathematics .. .'. All arguably true. And all statements obtainable this way form part of the raison d 'e1re of this series.

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Title:Measures and Differential Equations in Infinite-Dimensional SpaceFormat:PaperbackDimensions:337 pagesPublished:October 26, 2012Publisher:Springer-Verlag/Sci-Tech/TradeLanguage:English

The following ISBNs are associated with this title:

ISBN - 10:9401051488

ISBN - 13:9789401051484

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Table of Contents

I. Measures and quasimeasures. Integration.- 1. Realvalued measures on algebras of sets.- 1.1. Premeasures.- 1.2. Same tests for ?-additivity of premeasures.- 1.3. Measurable and topological Radon spaces.- 1.4. Cylindrical measures.- 2. Cylinder sets and cylindrical functions.- 2.1. General definition of cylinder set.- 2.2. Cylinder sets in a linear space X.- 2.3. Measurable linear space.- 2.4. Cylindrical functions.- 3. Quasimeasures. Integration.- 3.1. Quasimeasures.- 3.2. Integral with respect to a quasimeasure.- 3.3. Quasimeasures in a measurable linear space.- 3.4. Positive quasimeasures.- 3.5. Integration of noncylindrical functions.- 4. Supplement: Some notions related to the topology of linear spaces.- 4.1. Prenorms.- 4.2. Locally convex spaces.- 4.3. Duality of linear spaces.- 4.4. Rigged Hilbert spaces.- 4.5. Polars.- 4.6. Nuclear topology.- 4.7. Compactness.- 5. Chapter I: Supplementary remarks and historical comments.- II. Gaussian measures in Hilbert space.- 1. Gaussian measures in finite-dimensional spaces.- 1.1. Characteristic functional and density.- 1.2. Computation of certain integrals.- 1.3. Integration by parts.- 1.4. Solution of the Cauchy problem.- 2. Gaussian measures in Hilbert space.- 2.1. ?-additivity for a Gaussian cylindrical measure.- 2.2. Some transformations of Gaussian measures in X.- 2.3. Computation of integrals.- 2.4. Gaussian cylindrical measures with arbitrary correlation operator.- 3. Measurable linear functionals and operators.- 3.1. Measurable linear functionals.- 3.2. Measurable linear operators.- 3.3. Integration by parts.- 3.4. Expansion into orthogonal polynomials.- 4. Absolute continuity of Gaussian measures.- 4.1. Equivalence of measures in a product space.- 4.2. Equivalence of Gaussian measures which differ by their means.- 4.3. Equivalence of Gaussian measures with distinct correlation operators.- 4.4. Absolute continuity of measures obtained from Gaussian measures by certain transformations of space.- 5. Fourier-Wiener transformation.- 5.1. Fourier transformation with respect to a Gaussian measure.- 5.2. Fourier-Wiener transformation of entire nmctions.- 5.3. Connection between the Fourier-Wiener transformation and orthogonal polynomials.- 6. Complexvalued Gaussian quasimeasures.- 6.1. Feynman integrals.- 6.2. Integration of analytic functionals.- 6.3. Computation of certain Feynman integrals.- 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.- 1.1. Sufficient conditions for ?-additivity. Strong regularity.- 1.2. Necessary conditions for ?-additivityM.- 1.3. The Hilbert space case.- 1.4. Integral representations of the group of unitary operators.- 1.5. Continuous cylindrical measures.- 2. Sequences of Radon measures.- 2.1. Weak compaetness in a spaee of measures.- 2.2. Weak completeness of spaees of measures.- 2.3. Properties of R-spaces.- 2.4. Examples of R-spaces.- 2.5. Weak compaetness of a family of measures in a space X'.- 3. Chapter III: Supplementary remarks and historical comments.- IV. Differentiable measures and distributions.- 1. Differentiable functions, differentiable expressions.- 1.1. Derivatives of a vector function.- 1.2. Higher order derivatives.- 1.3. Linear differential expressions.- 1.4. Symmetrie and dissipative differential operators.- 2. Differentiable measures.- 2.1. Derivative of a measure.- 2.2. The logarithmie derivative.- 2.3. The derivative of a measure as an element of the dual space.- 2.4. Higher order derivatives.- 3. Distributions and generalized functions.- 3.1. Test functions and measures.- 3.2. Distributions. Operations on distributions.- 3.3. Generalized funetions and kernels.- 3.4. Fourier transformation of distributions.- 3.5. Differential expressions for distributions.- 4. Positive definiteness. Quasi-invariant distributions and bidistributions.- 4.1. Positive distributions.- 4.2. Integral representations of invariant generalized kernels.- 4.3. Quasi-invariant distributions.- 4.4. Integral representations for quasi-invariant bidistributions.- 5. Chapter IV: Supplementary remarks and historical comments.- V. Evolution differential equations.- 1. Weak solutions of evolution equations.- 1.1. Fundamental distributions.- 1.2. Systems of equations.- 1.3. Equations with constant operator.- 1.4. Operators with space-homogeneous symbol.- 1.5. Second order equations related to Gaussian measures.- 2. Second order equations with variable coefficient.- 2.1. Statement of the problem.- 2.2. Dimension independent apriori estimates.- 2.3. A priori estimates for derivatives of solutions of the Cauchy problem, independent of the number of arguments.- 2.4. Equations with cylindrical coefficients.- 2.5. Measures determined by sets of finite-dimensional equations.- 2.6. Solution of the Cauchy problem for a single equation in an infinitedimensional space.- 2.7. Systems of equations in an infinite-dimensional space.- 3. Chapter V: Supplementary remarks and historical comments.- VI. Integration in path space.- 1. Markov quasimeasures.- 1.1. Transition measures. A boundedness condition for a Markov quasimeasure.- 1.2. Integration with respect to a Markov quasimeasure.- 2. Evolution families of operators.- 2.1. Construction of the evolution family.- 2.2. The chronological product of evolution families.- 2.3. Linear evolution families with constant generating operator.- 2.4. Linear evolution families with variable generating operator.- 2.5. Evolution families related to quasilinear equations.- 3. Linear evolution families and functional integrals.- 3.1. Transition measures and evolution families. Integration of multiplicative functionals.- 3.2. Some particular cases.- 4. Nonlinear evolution families, and integrals in branching path space.- 4.1. Branching paths.- 4.2. Evolution families in the dass of formal power series.- 4.3. Construction of a multiplicative integral.- 4.4. Convergence of formal expansions.- 4.5. Markov quasimeasures and integrals along branching paths.- 5. Chapter VI: Supplementary remarks and historical comments.- VII. Probabilistic representations of solutions of parabolic equations and systems.- 1. Wiener process. Stochastic integrals.- 1.1. Basic not ions from the theory of stochastic processes.- 1.2. Stochastic integrals.- 1.3. Itô's formula.- 2. Stochastic differential equations.- 2.1. Equations with bounded operators.- 2.2. Equations with unbounded operator.- 2.3. Equations with random coefficients.- 3. Operator multiplicative functionals and the evolution families determined by them.- 3.1. Basic definitions.- 3.2. Ordinary linear equations.- 3.3. Stochastic linear equations.- 3.4. Smoothness with respect to the initial value of the solution of a stochastic differential equation and of the corresponding multiplicative functional.- 3.5. Invariance of smooth functions under an evolution family.- 4. The Cauchy problem for second order parabolic systems.- 4.1. The Cauchy problem for a single linear equation.- 4.2. The Cauchy problem for a linear system.- 4.3. Quasilinear systems.- 5. Chapter VII: Supplementary remarks and historical comments.- VIII. Smooth measures.- 1. Admissible operators.- 1.1. Definition and properties.- 1.2. Admissible operators defined by differential equations.- 2. Admissibility of differential operators.- 2.1. The logarithmic derivative of a measure along a vector field.- 2.2. Differential operators of higher order.- 3. Absolute continuity of smooth measures.- 3.1. The shift of a measure along a vector field.- 3.2. Equivalence of solutions of the forward Kolmogorov equation.- 3.3. Absolute continuity of measures under non linear transformations.- 4. Maps of spaces and admissible operators.- 4.1. Compatible pairs of operators.- 4.2. Smoothness of finite-dimensional images of measures.- 4.3. The quadratic map of smooth measures.- 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.