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This volume is to pique the interest of many researchers in the
fields of infinite dimensional analysis and quantum probability.
These fields have undergone increasingly significant developments
and have found many new applications, in particular, to classical
probability and to different branches of physics. These fields are
rather wide and are of a strongly interdisciplinary nature. For
such a purpose, we strove to bridge among these interdisciplinary
fields in our Workshop on IDAQP and their Applications that was
held at the Institute for Mathematical Sciences, National
University of Singapore from 3-7 March 2014. Readers will find that
this volume contains all the exciting contributions by well-known
researchers in search of new directions in these fields.
Why should we use white noise analysis? Well, one reason of course
is that it fills that earlier gap in the tool kit. As Hida would
put it, white noise provides us with a useful set of independent
coordinates, parametrized by 'time'. And there is a feature which
makes white noise analysis extremely user-friendly. Typically the
physicist - and not only he - sits there with some heuristic
ansatz, like e.g. the famous Feynman 'integral', wondering whether
and how this might make sense mathematically. In many cases the
characterization theorem of white noise analysis provides the user
with a sweet and easy answer. Feynman's 'integral' can now be
understood, the 'It's all in the vacuum' ansatz of Haag and Coester
is now making sense via Dirichlet forms, and so on in many fields
of application. There is mathematical finance, there have been
applications in biology, and engineering, many more than we could
collect in the present volume.Finally, there is one extra benefit:
when we internalize the structures of Gaussian white noise analysis
we will be ready to meet another close relative. We will enjoy the
important similarities and differences which we encounter in the
Poisson case, championed in particular by Y Kondratiev and his
group. Let us look forward to a companion volume on the uses of
Poisson white noise.The present volume is more than a collection of
autonomous contributions. The introductory chapter on white noise
analysis was made available to the other authors early on for
reference and to facilitate conceptual and notational coherence in
their work.
During the last fifty years, Gopinath Kallianpur has made extensive
and significant contributions to diverse areas of probability and
statistics, including stochastic finance, Fisher consistent
estimation, non-linear prediction and filtering problems, zero-one
laws for Gaussian processes and reproducing kernel Hilbert space
theory, and stochastic differential equations in infinite
dimensions. To honor Kallianpur's pioneering work and scholarly
achievements, a number of leading experts have written research
articles highlighting progress and new directions of research in
these and related areas. This commemorative volume, dedicated to
Kallianpur on the occasion of his seventy-fifth birthday, will pay
tribute to his multi-faceted achievements and to the deep insight
and inspiration he has so graciously offered his students and
colleagues throughout his career. Contributors to the volume: S.
Aida, N. Asai, K. B. Athreya, R. N. Bhattacharya, A. Budhiraja, P.
S. Chakraborty, P. Del Moral, R. Elliott, L. Gawarecki, D. Goswami,
Y. Hu, J. Jacod, G. W. Johnson, L. Johnson, T. Koski, N. V. Krylov,
I. Kubo, H.-H. Kuo, T. G. Kurtz, H. J. Kushner, V. Mandrekar, B.
Margolius, R. Mikulevicius, I. Mitoma, H. Nagai, Y. Ogura, K. R.
Parthasarathy, V. Perez-Abreu, E. Platen, B. V. Rao, B. Rozovskii,
I. Shigekawa, K. B. Sinha, P. Sundar, M. Tomisaki, M. Tsuchiya, C.
Tudor, W. A. Woycynski, J. Xiong
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.
During the last fifty years, Gopinath Kallianpur has made extensive
and significant contributions to diverse areas of probability and
statistics, including stochastic finance, Fisher consistent
estimation, non-linear prediction and filtering problems, zero-one
laws for Gaussian processes and reproducing kernel Hilbert space
theory, and stochastic differential equations in infinite
dimensions. To honor Kallianpur's pioneering work and scholarly
achievements, a number of leading experts have written research
articles highlighting progress and new directions of research in
these and related areas. This commemorative volume, dedicated to
Kallianpur on the occasion of his seventy-fifth birthday, will pay
tribute to his multi-faceted achievements and to the deep insight
and inspiration he has so graciously offered his students and
colleagues throughout his career. Contributors to the volume: S.
Aida, N. Asai, K. B. Athreya, R. N. Bhattacharya, A. Budhiraja, P.
S. Chakraborty, P. Del Moral, R. Elliott, L. Gawarecki, D. Goswami,
Y. Hu, J. Jacod, G. W. Johnson, L. Johnson, T. Koski, N. V. Krylov,
I. Kubo, H.-H. Kuo, T. G. Kurtz, H. J. Kushner, V. Mandrekar, B.
Margolius, R. Mikulevicius, I. Mitoma, H. Nagai, Y. Ogura, K. R.
Parthasarathy, V. Perez-Abreu, E. Platen, B. V. Rao, B. Rozovskii,
I. Shigekawa, K. B. Sinha, P. Sundar, M. Tomisaki, M. Tsuchiya, C.
Tudor, W. A. Woycynski, J. Xiong
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.
The use of probabilistic methods in the biological sciences has
been so well established by now that mathematical biology is
regarded by many as a distinct dis cipline with its own repertoire
of techniques. The purpose of the Workshop on sto chastic methods
in biology held at Nagoya University during the week of July 8-12,
1985, was to enable biologists and probabilists from Japan and the
U. S. to discuss the latest developments in their respective fields
and to exchange ideas on the ap plicability of the more recent
developments in stochastic process theory to problems in biology.
Eighteen papers were presented at the Workshop and have been
grouped under the following headings: I. Population genetics (five
papers) II. Measure valued diffusion processes related to
population genetics (three papers) III. Neurophysiology (two
papers) IV. Fluctuation in living cells (two papers) V.
Mathematical methods related to other problems in biology,
epidemiology, population dynamics, etc. (six papers) An important
feature of the Workshop and one of the reasons for organizing it
has been the fact that the theory of stochastic differential
equations (SDE's) has found a rich source of new problems in the
fields of population genetics and neuro biology. This is especially
so for the relatively new and growing area of infinite dimensional,
i. e., measure-valued or distribution-valued SDE's. The papers in
II and III and some of the papers in the remaining categories
represent these areas."
Encompassing both introductory and more advanced research material,
these notes deal with the author's contributions to stochastic
processes and focus on Brownian motion processes and its derivative
white noise. Originally published in 1970. The Princeton Legacy
Library uses the latest print-on-demand technology to again make
available previously out-of-print books from the distinguished
backlist of Princeton University Press. These editions preserve the
original texts of these important books while presenting them in
durable paperback and hardcover editions. The goal of the Princeton
Legacy Library is to vastly increase access to the rich scholarly
heritage found in the thousands of books published by Princeton
University Press since its founding in 1905.
Encompassing both introductory and more advanced research material,
these notes deal with the author's contributions to stochastic
processes and focus on Brownian motion processes and its derivative
white noise. Originally published in 1970. The Princeton Legacy
Library uses the latest print-on-demand technology to again make
available previously out-of-print books from the distinguished
backlist of Princeton University Press. These editions preserve the
original texts of these important books while presenting them in
durable paperback and hardcover editions. The goal of the Princeton
Legacy Library is to vastly increase access to the rich scholarly
heritage found in the thousands of books published by Princeton
University Press since its founding in 1905.
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