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Record Nr. |
UNINA9910810359903321 |
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Autore |
Hida Takeyuki <1927-> |
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Titolo |
Lectures on white noise functionals / / T. Hida, Si Si |
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Pubbl/distr/stampa |
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Hackensack, NJ, : World Scientific, c2008 |
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ISBN |
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Edizione |
[1st ed.] |
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Descrizione fisica |
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1 online resource (280 p.) |
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Altri autori (Persone) |
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Disciplina |
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Soggetti |
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White noise theory |
Gaussian processes |
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Lingua di pubblicazione |
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Formato |
Materiale a stampa |
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Livello bibliografico |
Monografia |
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Note generali |
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Description based upon print version of record. |
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Nota di bibliografia |
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Includes bibliographical references (p. 253-261) and index. |
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Nota di contenuto |
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Preface; Contents; 1. Introduction; 1.1 Preliminaries; 1.2 Our idea of establishing white noise analysis; 1.3 A brief synopsis of the book; 1.4 Some general background; 1.4.1 Characteristics of white noise analysis; 2. Generalized white noise functionals; 2.1 Brownian motion and Poisson process; elemental stochastic processes; 2.2 Comparison between Brownian motion and Poisson process; 2.3 The Bochner-Minlos theorem; 2.4 Observation of white noise through the L evy's construction of Brownian motion; 2.5 Spaces (L2), F and F arising from white noise; 2.6 Generalized white noise functionals |
A. Use of the Sobolev space structureB. An analogue of the Schwartz space.; 2.7 Creation and annihilation operators; 2.8 Examples; 2.9 Addenda; A.1. The Gauss transform, the S-transform and applications; A.2. The Karhunen-Lo eve expansion; A.3. Reproducing kernel Hilbert space; 3. Elemental random variables and Gaussian processes; 3.1 Elemental noises; I. The first method of stochastic integral.; II. The second method of stochastic integral.; 3.2 Canonical representation of a Gaussian process; 3.3 Multiple Markov Gaussian processes; 3.4 Fractional Brownian motion |
3.5 Stationarity of fractional Brownian motion3.6 Fractional order differential operator in connection with L evy's Brownian motion; 3.7 Gaussian random fields; 4. Linear processes and linear fields; 4.1 Gaussian systems; 4.2 Poisson systems; 4.3 Linear functionals of Poisson noise; 4.4 Linear processes; 4.5 L evy field and generalized L evy field; 4.6 Gaussian elemental noises; 5. Harmonic analysis arising |
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from infinite dimensional rotation group; 5.1 Introduction; 5.2 Infinite dimensional rotation group O(E); 5.3 Harmonic analysis; 5.4 Addenda to the diagram |
5.5 The L evy group, the Windmill subgroup and the sign-changing subgroup of O(E)5.6 Classification of rotations in O(E); 5.7 Unitary representation of the infinite dimensional rotation group O(E); 5.8 Laplacian; 6. Complex white noise and infinite dimensional unitary group; 6.1 Why complex?; 6.2 Some background; 6.3 Subgroups of U(Ec); 6.4 Applications; I. Symmetry of the heat equation and the Schr odinger equation.; II. Analysis on half plane of E; 7. Characterization of Poisson noise; 7.1 Preliminaries; 7.2 A characteristic of Poisson noise; 7.3 A characterization of Poisson noise |
7.4 Comparison of two noises Gaussian and Poisson; 7.5 Poisson noise functionals; 8. Innovation theory; 8.1 A short history of innovation theory; 8.2 Definitions and examples; 8.3 Innovations in the weak sense; 8.4 Some other concrete examples; 9. Variational calculus for random fields and operator fields; 9.1 Introduction; 9.2 Stochastic variational equations; 9.3 Illustrative examples; 9.4 Integrals of operators; 9.4.1 Operators of linear form; 9.4.2 Operators of quadratic forms of the creation and the annihilation operators; 9.4.3 Polynomials in R; of degree 2 |
10. Four notable roads to quantum dynamics |
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Sommario/riassunto |
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White noise analysis is an advanced stochastic calculus that has developed extensively since three decades ago. It has two main characteristics. One is the notion of generalized white noise functionals, the introduction of which is oriented by the line of advanced analysis, and they have made much contribution to the fields in science enormously. The other characteristic is that the white noise analysis has an aspect of infinite dimensional harmonic analysis arising from the infinite dimensional rotation group. With the help of this rotation group, the white noise analysis has explored new are |
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