LEADER 05329nam 2200649Ia 450 001 9910455529303321 005 20200520144314.0 010 $a981-281-204-0 035 $a(CKB)1000000000767707 035 $a(EBL)1193660 035 $a(SSID)ssj0000518839 035 $a(PQKBManifestationID)12187114 035 $a(PQKBTitleCode)TC0000518839 035 $a(PQKBWorkID)10494530 035 $a(PQKB)11653555 035 $a(MiAaPQ)EBC1193660 035 $a(WSP)00001447 035 $a(PPN)181396025 035 $a(Au-PeEL)EBL1193660 035 $a(CaPaEBR)ebr10688114 035 $a(CaONFJC)MIL491719 035 $a(OCoLC)820944509 035 $a(EXLCZ)991000000000767707 100 $a20080125d2008 uy 0 101 0 $aeng 135 $aur|n|---||||| 181 $ctxt 182 $cc 183 $acr 200 10$aLectures on white noise functionals$b[electronic resource] /$fT. Hida, Si Si 210 $aHackensack, NJ $cWorld Scientific$dc2008 215 $a1 online resource (280 p.) 300 $aDescription based upon print version of record. 311 $a981-256-052-1 320 $aIncludes bibliographical references (p. 253-261) and index. 327 $aPreface; 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 327 $aA. 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 327 $a3.5 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 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 327 $a5.5 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 327 $a7.4 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 2 327 $a10. Four notable roads to quantum dynamics 330 $aWhite 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 606 $aWhite noise theory 606 $aGaussian processes 608 $aElectronic books. 615 0$aWhite noise theory. 615 0$aGaussian processes. 676 $a519.2/2 700 $aHida$b Takeyuki$f1927-$047700 701 $aSi$b Si$0868938 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 906 $aBOOK 912 $a9910455529303321 996 $aLectures on white noise functionals$92172428 997 $aUNINA