Measures and Differential Equations in Infinite-Dimensional Space by Yu.L. DaleckyMeasures and Differential Equations in Infinite-Dimensional Space by Yu.L. Dalecky

Measures and Differential Equations in Infinite-Dimensional Space

byYu.L. Dalecky, S.V. Fomin

Paperback | October 26, 2012

Pricing and Purchase Info

$156.70 online 
$165.95 list price save 5%
Earn 784 plum® points

Prices and offers may vary in store

Quantity:

In stock online

Ships free on orders over $25

Not available in stores

about

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.
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

Reviews

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.