LEADER 04654nam 2200661 a 450 001 9910821509903321 005 20240516111604.0 010 $a1-280-36188-3 010 $a9786613555250 010 $a981-283-689-6 035 $a(CKB)2550000000079601 035 $a(EBL)840715 035 $a(OCoLC)858228489 035 $a(SSID)ssj0000599352 035 $a(PQKBManifestationID)12232702 035 $a(PQKBTitleCode)TC0000599352 035 $a(PQKBWorkID)10596340 035 $a(PQKB)10897660 035 $a(MiAaPQ)EBC840715 035 $a(WSP)00007103 035 $a(Au-PeEL)EBL840715 035 $a(CaPaEBR)ebr10524572 035 $a(CaONFJC)MIL355525 035 $a(EXLCZ)992550000000079601 100 $a20120123d2012 uy 0 101 0 $aeng 135 $aur|n|---||||| 181 $ctxt 182 $cc 183 $acr 200 10$aIntroduction to Hida distributions /$fSi Si 205 $a1st ed. 210 $aSingapore ;$aHackensack, N.J. $cWorld Scientific$dc2012 215 $a1 online resource (268 p.) 300 $aDescription based upon print version of record. 311 $a981-283-688-8 320 $aIncludes bibliographical references (p. 243-249) and index. 327 $aPreface; Contents; 1. Preliminaries and Discrete Parameter White Noise; 1.1 Preliminaries; 1.2 Discrete parameter white noise; 1.3 Invariance of the measure ?; 1.4 Harmonic analysis arising from O(E) on the space of functionals of Y = {Y (n)}; 1.5 Quadratic forms; 1.6 Differential operators and related operators; 1.7 Probability distributions and Bochner-Minlos theorem; 2. Continuous Parameter White Noise; 2.1 Gaussian system; 2.2 Continuous parameter white noise; 2.3 Characteristic functional and Bochner-Minlos theorem; 2.4 Passage from discrete to continuous 327 $a2.5 Stationary generalized stochastic processes3. White Noise Functionals; 3.1 In line with standard analysis; 3.2 White noise functionals; 3.3 Infinite dimensional spaces spanned by generalized linear functionals of white noise; 3.4 Some of the details of quadratic functionals of white noise; 3.5 The T -transform and the S-transform; 3.6 White noise (t) related to ?-function; 3.7 Infinite dimensional space generated by Hermite polynomials in (t)'s of higher degree; 3.8 Generalized white noise functionals; 3.9 Approximation to Hida distributions 327 $a3.10 Renormalization in Hida distribution theory4. White Noise Analysis; 4.1 Operators acting on (L2)-; 4.2 Application to stochastic differential equation; 4.3 Differential calculus and Laplacian operators; 4.4 Infinite dimensional rotation group O(E); 4.5 Addenda; 5. Stochastic Integral; 5.1 Introduction; 5.2 Wiener integrals and multiple Wiener integrals; 5.3 The Ito integral; 5.4 Hitsuda-Skorokhod integrals; 5.5 Levy's stochastic integral; 5.6 Addendum : Path integrals; 6. Gaussian and Poisson Noises; 6.1 Poisson noise and its probability distribution 327 $a6.2 Comparison between the Gaussian white noise and the Poisson noise, with the help of characterization of measures6.3 Symmetric group in Poisson noise analysis; 6.4 Spaces of quadratic Hida distributions and their dualities; 7. Multiple Markov Properties of Generalized Gaussian Processes and Generalizations; 7.1 A brief discussion on canonical representation theory for Gaussian processes and multiple Markov property; 7.2 Duality for multiple Markov Gaussian processes in the restricted sense; 7.3 Uniformly multiple Markov processes 327 $a9.3 Stable distribution 330 $aThis book provides the mathematical definition of white noise and gives its significance. White noise is in fact a typical class of idealized elemental (infinitesimal) random variables. Thus, we are naturally led to have functionals of such elemental random variables that is white noise. This book analyzes those functionals of white noise, particularly the generalized ones called Hida distributions, and highlights some interesting future directions. The main part of the book involves infinite dimensional differential and integral calculus based on the variable which is white noise. The present 606 $aWhite noise theory 606 $aStochastic analysis 606 $aStochastic differential equations 615 0$aWhite noise theory. 615 0$aStochastic analysis. 615 0$aStochastic differential equations. 676 $a519.22 700 $aSi$b Si$01662707 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 906 $aBOOK 912 $a9910821509903321 996 $aIntroduction to Hida distributions$94024385 997 $aUNINA