Lectures on white noise functionals /:
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 c...
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| Other Authors: | |
| Format: | Electronic eBook |
| Language: | English |
| Published: |
Singapore ; Hackensack, N.J. :
World Scientific Pub. Co.,
©2008.
|
| Subjects: | |
| Online Access: | DE-862 DE-863 |
| Summary: | 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 areas of mathematics and has extended the fields of applications. |
| Physical Description: | 1 online resource |
| Bibliography: | Includes bibliographical references (pages 253-261) and index. |
| ISBN: | 9789812812049 9812812040 |
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| 505 | 0 | |a 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 -- 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 Lévy's construction of Brownian motion. 2.5. Spaces [symbol], F and [symbol] arising from white noise. 2.6. Generalized white noise functionals. 2.7. Creation and annihilation operators. 2.8. Examples. 2.9. Addenda. | |
| 505 | 0 | |a 3. Elemental random variables and Gaussian processes. 3.1. Elemental noises. 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 motion. 3.6. Fractional order differential operator in connection with Lévy'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. Lévy field and generalized Lévy field. 4.6. Gaussian elemental noises. | |
| 505 | 0 | |a 5. Harmonic analysis arising 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 Lévy 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 [symbol]. 6.4. Applications -- 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. | |
| 505 | 0 | |a 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 -- 10. Four notable roads to quantum dynamics. 10.1. White noise approach to path integrals. 10.2. Hamiltonian dynamics and Chern-Simons functional integrals. 10.3. Dirichlet forms. 10.4. Time operator. 10.5. Addendum: Euclidean fields. | |
| 520 | |a 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 areas of mathematics and has extended the fields of applications. | ||
| 650 | 0 | |a White noise theory. |0 http://id.loc.gov/authorities/subjects/sh2007010827 | |
| 650 | 0 | |a Gaussian processes. |0 http://id.loc.gov/authorities/subjects/sh85053558 | |
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| 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 -- 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 Lévy's construction of Brownian motion. 2.5. Spaces [symbol], F and [symbol] arising from white noise. 2.6. Generalized white noise functionals. 2.7. Creation and annihilation operators. 2.8. Examples. 2.9. Addenda. 3. Elemental random variables and Gaussian processes. 3.1. Elemental noises. 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 motion. 3.6. Fractional order differential operator in connection with Lévy'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. Lévy field and generalized Lévy field. 4.6. Gaussian elemental noises. 5. Harmonic analysis arising 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 Lévy 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 [symbol]. 6.4. Applications -- 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 -- 10. Four notable roads to quantum dynamics. 10.1. White noise approach to path integrals. 10.2. Hamiltonian dynamics and Chern-Simons functional integrals. 10.3. Dirichlet forms. 10.4. Time operator. 10.5. Addendum: Euclidean fields. |
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| spelling | Hida, Takeyuki, 1927-2017. https://id.oclc.org/worldcat/entity/E39PBJdgYvKWHh38GbT8JwkhpP Lectures on white noise functionals / by T. Hida & Si Si. Singapore ; Hackensack, N.J. : World Scientific Pub. Co., ©2008. 1 online resource text txt rdacontent computer c rdamedia online resource cr rdacarrier Includes bibliographical references (pages 253-261) and index. 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 -- 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 Lévy's construction of Brownian motion. 2.5. Spaces [symbol], F and [symbol] arising from white noise. 2.6. Generalized white noise functionals. 2.7. Creation and annihilation operators. 2.8. Examples. 2.9. Addenda. 3. Elemental random variables and Gaussian processes. 3.1. Elemental noises. 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 motion. 3.6. Fractional order differential operator in connection with Lévy'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. Lévy field and generalized Lévy field. 4.6. Gaussian elemental noises. 5. Harmonic analysis arising 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 Lévy 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 [symbol]. 6.4. Applications -- 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 -- 10. Four notable roads to quantum dynamics. 10.1. White noise approach to path integrals. 10.2. Hamiltonian dynamics and Chern-Simons functional integrals. 10.3. Dirichlet forms. 10.4. Time operator. 10.5. Addendum: Euclidean fields. 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 areas of mathematics and has extended the fields of applications. White noise theory. http://id.loc.gov/authorities/subjects/sh2007010827 Gaussian processes. http://id.loc.gov/authorities/subjects/sh85053558 Théorie du bruit blanc. Processus gaussiens. MATHEMATICS Probability & Statistics General. bisacsh Gaussian processes fast White noise theory fast Electronic books. Si, Si. |
| spellingShingle | Hida, Takeyuki, 1927-2017 Lectures on white noise functionals / 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 -- 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 Lévy's construction of Brownian motion. 2.5. Spaces [symbol], F and [symbol] arising from white noise. 2.6. Generalized white noise functionals. 2.7. Creation and annihilation operators. 2.8. Examples. 2.9. Addenda. 3. Elemental random variables and Gaussian processes. 3.1. Elemental noises. 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 motion. 3.6. Fractional order differential operator in connection with Lévy'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. Lévy field and generalized Lévy field. 4.6. Gaussian elemental noises. 5. Harmonic analysis arising 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 Lévy 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 [symbol]. 6.4. Applications -- 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 -- 10. Four notable roads to quantum dynamics. 10.1. White noise approach to path integrals. 10.2. Hamiltonian dynamics and Chern-Simons functional integrals. 10.3. Dirichlet forms. 10.4. Time operator. 10.5. Addendum: Euclidean fields. White noise theory. http://id.loc.gov/authorities/subjects/sh2007010827 Gaussian processes. http://id.loc.gov/authorities/subjects/sh85053558 Théorie du bruit blanc. Processus gaussiens. MATHEMATICS Probability & Statistics General. bisacsh Gaussian processes fast White noise theory fast |
| subject_GND | http://id.loc.gov/authorities/subjects/sh2007010827 http://id.loc.gov/authorities/subjects/sh85053558 |
| title | Lectures on white noise functionals / |
| title_auth | Lectures on white noise functionals / |
| title_exact_search | Lectures on white noise functionals / |
| title_full | Lectures on white noise functionals / by T. Hida & Si Si. |
| title_fullStr | Lectures on white noise functionals / by T. Hida & Si Si. |
| title_full_unstemmed | Lectures on white noise functionals / by T. Hida & Si Si. |
| title_short | Lectures on white noise functionals / |
| title_sort | lectures on white noise functionals |
| topic | White noise theory. http://id.loc.gov/authorities/subjects/sh2007010827 Gaussian processes. http://id.loc.gov/authorities/subjects/sh85053558 Théorie du bruit blanc. Processus gaussiens. MATHEMATICS Probability & Statistics General. bisacsh Gaussian processes fast White noise theory fast |
| topic_facet | White noise theory. Gaussian processes. Théorie du bruit blanc. Processus gaussiens. MATHEMATICS Probability & Statistics General. Gaussian processes White noise theory Electronic books. |
| work_keys_str_mv | AT hidatakeyuki lecturesonwhitenoisefunctionals AT sisi lecturesonwhitenoisefunctionals |