White Noise: An Infinite Dimensional CalculusSpringer Science & Business Media, 29 ביוני 2013 - 520 עמודים Many areas of applied mathematics call for an efficient calculus in infinite dimensions. This is most apparent in quantum physics and in all disciplines of science which describe natural phenomena by equations involving stochasticity. With this monograph we intend to provide a framework for analysis in infinite dimensions which is flexible enough to be applicable in many areas, and which on the other hand is intuitive and efficient. Whether or not we achieved our aim must be left to the judgment of the reader. This book treats the theory and applications of analysis and functional analysis in infinite dimensions based on white noise. By white noise we mean the generalized Gaussian process which is (informally) given by the time derivative of the Wiener process, i.e., by the velocity of Brownian mdtion. Therefore, in essence we present analysis on a Gaussian space, and applications to various areas of sClence. Calculus, analysis, and functional analysis in infinite dimensions (or dimension-free formulations of these parts of classical mathematics) have a long history. Early examples can be found in the works of Dirichlet, Euler, Hamilton, Lagrange, and Riemann on variational problems. At the beginning of this century, Frechet, Gateaux and Volterra made essential contributions to the calculus of functions over infinite dimensional spaces. The important and inspiring work of Wiener and Levy followed during the first half of this century. Moreover, the articles and books of Wiener and Levy had a view towards probability theory. |
תוכן
1 | |
Generalized Functionals | 35 |
B Triples of Functional Spaces | 52 |
The Spaces 9 and | 74 |
B Positive Generalized Functionals | 109 |
Calculus of Differential Operators | 146 |
Laplacian Operators | 184 |
E The Commutation Relations | 228 |
Fourier and FourierMehler Transforms | 317 |
E A Characterization of the Fourier Transform | 336 |
H FourierMehler Transforms | 356 |
B Closability and the Associated Diffusion Processes | 382 |
Applications to Quantum Field Theory | 399 |
Feynman Integrals | 435 |
Appendices | 451 |
Reproducing Kernel Hilbert Spaces | 468 |
The Spaces I and I | 232 |
B LittlewoodPaleyStein Inequalities | 249 |
Stochastic Integration | 277 |
Bibliography | 486 |
Notations and Conventions | 507 |
מהדורות אחרות - הצג הכל
White Noise: An Infinite Dimensional Calculus <span dir=ltr>Takeyuki Hida</span> תצוגה מקדימה מוגבלת - 1993 |
White Noise: An Infinite Dimensional Calculus <span dir=ltr>Takeyuki Hida</span> אין תצוגה מקדימה זמינה - 1993 |
מונחים וביטויים נפוצים
adjoint Albeverio algebra Assume bounded Brownian motion calculus chaos decomposition Chapter closable compute consider construct continuous linear operator converges Corollary definition denote densely defined Dirichlet forms dual dµ(x element in 9 equation ES(R estimate Example exists EY(R Feynman finite Fock space following result formula Fourier transform Fréchet Gâteaux derivative Gâteaux differentiable Gaussian measure given Hermite functions Hida Hilbert space Høegh-Krohn implies inequality infinite dimensional Itô kernel Kubo L²(R L²(v Laplacian Lemma Malliavin calculus mapping Math Moreover norm notation Note number operator obtain operator from 9 pointwise polynomials proof of Theorem Proposition prove quantum field theory random variables right hand side Röckner S-transform scalar product Schwartz Schwinger functions self-adjoint semigroup strongly continuous subspace symmetric topology U-functional vector spaces white noise white noise analysis white noise functionals zero αμ