Parter-Wiener, etc. Theory

Frederik Jozef Belinfante

R. F. Streater

Norbert Wiener Centenary Congress, Vidyadhar S. Mandrekar, P. R. Masani

Wiener-Kolmogorov conception of the stochastic organization of nature by S. Molchanov The Wiener programme in statistical physics. Is it feasible in the light of recent developments? by O. Penrose The mathematical ramifications of Wiener's program in statistical physics by L. Gross Potential capacity before and after Wiener by D. R. Adams Generalized harmonic analysis and Gabor and wavelet systems by J. J. Benedetto Wiener-Hopf equations and linear systems by J. Bart, I. Gohberg, and M. A. Kaashoek Complex harmonic analysis in the aftermath of Paley-Wiener by P. Malliavin Amalgam spaces and generalized harmonic analysis by H. G. Feichtinger Wiener space and nonlinear quantum field theory by I. Segal Wiener and Feynman--path integrals and their applications by S. Albeverio Optical coherence before and after Wiener by J. R. Klauder Wiener and the problem of Hidden Parameters in quantum mechanics by E. A. Carlen Finite path integrals by E. F. Thomas Joint probabilities of noncommuting observables and the Einstein-Podolsky-Rosen question in Wiener-Siegel quantum theory by R. L. Warnock The role of Leibniz and Haldane in Wiener's cybernetics by G. Gale Quantum mechanical coherence, resonance, and mind by H. P. Stapp What is mind that the brain may order it? by K. H. Pribram Shannon-Wiener information and stochastic complexity by J. Rissanen Sharp norm comparison of Martingale maximal functions and stochastic integrals by D. L. Burkholder Stochastic analysis on Wiener space by S. Watanabe Stochastic filtering: A part of stochastic nonlinear analysis by G. Kallianpur The theory of learning before and after Wiener by N. K. Bose Sensory and motor prostheses in the aftermath of Wiener by R. W. Mann On Wiener's thought in the computer as an aid in visualizing higher-dimensioanl forms and its modern ramifications by D. Roseman Norbert Wiener and the future of cybernetics by P. R. Masani Letter to Norbert Wiener from John von Neumann Scientific impact of the work of C. E. Shannon by B. McMillan Citation and award of the Norbert Wiener Centenary Medal to Professor C. E. Shannon (in absentia) by J. Ramaekers Wiener and the art of communication by D. Gabor Academic vita of Norbert Wiener Doctoral students of Norbert Wiener Bibliography of Norbert Wiener Defense Department documents List of participants (Norbert Wiener centenary congress).

Mateusz Jurczynski

In this thesis we develop the theory of quantum Wiener integrals on the bosonic Fock space. We study multiple quantum Wiener integrals as an algebra of unbounded operators, investigating its properties, including closedness, common domains and multiplication formulas. We show the applications of the new formalism by providing new proofs to the established theory of quantum stochastic calculus and new conditions for generating quantum stochastic cocycles and quantum stochastic evolutions. The corresponding quasifree case is also studied and the constructions extended to fit in that formalism. We construct the multiple quantum Wiener integral as one operator on a family of operators which we dub operator kernels. This in particular covers the case of quantum stochastic cocycles and evolutions. We show that the family of quantum Wiener integrals forms a WOT-dense algebra of unbounded operators on the bosonic Fock space. We provide more general conditions for an operator kernel to be multiple quantum Wiener integrable, which allows us to treat multiple quantum Wiener integrals as an algebra. We explore the influence of an initial space on the theory. Our setting gives natural conditions for a product of two cocycles (evolutions) to still be a cocycle (an evolution). We apply our theory by solving quantum stochastic differential equations (QSDEs) and by finding more elementary proofs of structure conditions on the generator of a quantum stochastic evolution and of the fundamental estimate in the proof of quantum stochastic Lie–Trotter formula. We also show how our theory unifies and generalises the theory of integral kernels and chaotic representation properties, proving in particular that every Hilbert–Schmidt operator is a quantum Wiener integral.

Fabio Franchini

Dong Hyun Cho

In this article, we introduce an integral transform as operator-valued Feynman integrals over Wiener paths in abstract Wiener space. In addition, we evaluate the integral transform for various types of functions, which are of interest in quantum mechanics and Feynman integration theories, via the conditional Feynman integral over Wiener paths in abstract Wiener space as a kernel.