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"An Introduction to Quantum Stochastic Calculus" aims to deepen
our understanding of the dynamics of systems subject to the laws of
chance both from the classical and the quantum points of view and
stimulate further research in their unification. This is probably
the first systematic attempt to weave classical probability theory
into the quantum framework and provides a wealth of interesting
features:
The origin of Ito s correction formulae for Brownian motion and the
Poisson process can be traced to commutation relations or,
equivalently, the uncertainty principle.
Quantum stochastic integration enables the possibility of seeing
new relationships between fermion and boson fields.
Many quantum dynamical semigroups as well as classical Markov
semigroups are realised through unitary operator evolutions.
The text is almost self-contained and requires only an elementary
knowledge of operator theory and probability theory at the graduate
level.
- - -
"This is an excellent volume which will be a valuable companion
bothto those who are already active in the field and those who are
new to it. Furthermore there are a large number of stimulating
exercises scattered through the text which will be invaluable to
students."
(Mathematical Reviews)
"This monograph gives a systematic and self-contained introduction
to the Fock space quantum stochastic calculus in its basic form
(...) by making emphasis on the mathematical aspects of quantum
formalism and its connections with classical probability and by
extensive presentation of carefully selected functional analytic
material. This makes the book very convenient for a reader with the
probability-theoretic orientation, wishing to make acquaintance
with wonders of the noncommutative probability, and, more
specifcally, for a mathematics student studying this field."
(Zentralblatt MATH)
"Elegantly written, with obvious appreciation for fine points of
higher mathematics (...) most notable is the] author's effort to
weave classical probability theory into a] quantum framework.
"(The American Mathematical Monthly)
"Elegantly written, with obvious appreciation for fine points of
higher mathematics...most notable is [the] author's effort to weave
classical probability theory into [a] quantum framework." - The
American Mathematical Monthly "This is an excellent volume which
will be a valuable companion both for those who are already active
in the field and those who are new to it. Furthermore there are a
large number of stimulating exercises scattered through the text
which will be invaluable to students." - Mathematical Reviews An
Introduction to Quantum Stochastic Calculus aims to deepen our
understanding of the dynamics of systems subject to the laws of
chance both from the classical and the quantum points of view and
stimulate further research in their unification. This is probably
the first systematic attempt to weave classical probability theory
into the quantum framework and provides a wealth of interesting
features: The origin of Ito's correction formulae for Brownian
motion and the Poisson process can be traced to communication
relations or, equivalently, the uncertainty principle. Quantum
stochastic interpretation enables the possibility of seeing new
relationships between fermion and boson fields. Quantum dynamical
semigroups as well as classical Markov semigroups are realized
through unitary operator evolutions. The text is almost
self-contained and requires only an elementary knowledge of
operator theory and probability theory at the graduate level.
From the Preface: Srinivasa Varadhan began his research career at
the Indian Statistical Institute (ISI), Calcutta, where he started
as a graduate student in 1959. His first paper appeared in Sankhya,
the Indian Journal of Statistics in 1962. Together with his fellow
students V. S. Varadarajan, R. Ranga Rao and K. R. Parthasarathy,
Varadhan began the study of probability on topological groups and
on Hilbert spaces, and quickly gained an international reputation.
At this time Varadhan realised that there are strong connections
between Markov processes and differential equations, and in 1963 he
came to the Courant Institute in New York, where he has stayed ever
since. Here he began working with the probabilists Monroe Donsker
and Marc Kac, and a graduate student named Daniel Stroock. He wrote
a series of papers on the Martingale Problem and Diffusions
together with Stroock, and another series of papers on Large
Deviations together with Donsker. With this work Varadhan's
reputation as one of the leading mathematicians of the time was
firmly established. Since then he has contributed to several other
areas of probability, analysis and physics, and collaborated with
numerous distinguished mathematicians. Varadhan was awarded the
Abel Prize in 2007. These Collected Works contain all his research
papers over the half-century spanning 1962 to early 2012. Volume I
includes the introductory material, the papers on limit theorems
and review articles.
From the Preface: Srinivasa Varadhan began his research career at
the Indian Statistical Institute (ISI), Calcutta, where he started
as a graduate student in 1959. His first paper appeared in Sankhya,
the Indian Journal of Statistics in 1962. Together with his fellow
students V. S. Varadarajan, R. Ranga Rao and K. R. Parthasarathy,
Varadhan began the study of probability on topological groups and
on Hilbert spaces, and quickly gained an international reputation.
At this time Varadhan realised that there are strong connections
between Markov processes and differential equations, and in 1963 he
came to the Courant Institute in New York, where he has stayed ever
since. Here he began working with the probabilists Monroe Donsker
and Marc Kac, and a graduate student named Daniel Stroock. He wrote
a series of papers on the Martingale Problem and Diffusions
together with Stroock, and another series of papers on Large
Deviations together with Donsker. With this work Varadhan's
reputation as one of the leading mathematicians of the time was
firmly established. Since then he has contributed to several other
areas of probability, analysis and physics, and collaborated with
numerous distinguished mathematicians. Varadhan was awarded the
Abel Prize in 2007. These Collected Works contain all his research
papers over the half-century spanning 1962 to early 2012. Volume II
includes the papers on PDE, SDE, diffusions, and random media.
From the Preface: Srinivasa Varadhan began his research career at
the Indian Statistical Institute (ISI), Calcutta, where he started
as a graduate student in 1959. His first paper appeared in Sankhya,
the Indian Journal of Statistics in 1962. Together with his fellow
students V. S. Varadarajan, R. Ranga Rao and K. R. Parthasarathy,
Varadhan began the study of probability on topological groups and
on Hilbert spaces, and quickly gained an international reputation.
At this time Varadhan realised that there are strong connections
between Markov processes and differential equations, and in 1963 he
came to the Courant Institute in New York, where he has stayed ever
since. Here he began working with the probabilists Monroe Donsker
and Marc Kac, and a graduate student named Daniel Stroock. He wrote
a series of papers on the Martingale Problem and Diffusions
together with Stroock, and another series of papers on Large
Deviations together with Donsker. With this work Varadhan's
reputation as one of the leading mathematicians of the time was
firmly established. Since then he has contributed to several other
areas of probability, analysis and physics, and collaborated with
numerous distinguished mathematicians. Varadhan was awarded the
Abel Prize in 2007. These Collected Works contain all his research
papers over the half-century spanning 1962 to early 2012. Volume
III includes the papers on large deviations.
From the Preface: Srinivasa Varadhan began his research career at
the Indian Statistical Institute (ISI), Calcutta, where he started
as a graduate student in 1959. His first paper appeared in Sankhya,
the Indian Journal of Statistics in 1962. Together with his fellow
students V. S. Varadarajan, R. Ranga Rao and K. R. Parthasarathy,
Varadhan began the study of probability on topological groups and
on Hilbert spaces, and quickly gained an international reputation.
At this time Varadhan realised that there are strong connections
between Markov processes and differential equations, and in 1963 he
came to the Courant Institute in New York, where he has stayed ever
since. Here he began working with the probabilists Monroe Donsker
and Marc Kac, and a graduate student named Daniel Stroock. He wrote
a series of papers on the Martingale Problem and Diffusions
together with Stroock, and another series of papers on Large
Deviations together with Donsker. With this work Varadhan's
reputation as one of the leading mathematicians of the time was
firmly established. Since then he has contributed to several other
areas of probability, analysis and physics, and collaborated with
numerous distinguished mathematicians. Varadhan was awarded the
Abel Prize in 2007. These Collected Works contain all his research
papers over the half-century spanning 1962 to early 2012. Volume IV
includes the papers on particle systems.
The aim of this little book is to convey three principal
developments in the evolution of modern information theory:
Shannon's initiation of a revolution in 1948 by his interpretation
of Boltzmann entropy as a measure of information yielded by an
elementary statistical experiment and basic coding theorems on
storing messages and transmitting them through noisy communication
channels in an optimal manner; the influence of ergodic theory in
the enlargement of the scope of Shannon's theorems through the
works of McMillan, Feinstein, Wolfowitz, Breiman and others and its
impact on the appearance of the Kolmogorov-Sinai invariant for
elementary dynamical systems; and finally, the more recent work of
Schumacher, Holevo, Winter and others on the role of von Neumann
entropy in the quantum avatar of the basic coding theorems when
messages are encoded as quantum states, transmitted through noisy
quantum channels, and retrieved by generalized measurements.
The aim of this little book is to convey three principal
developments in the evolution of modern information theory:
Shannon's initiation of a revolution in 1948 by his interpretation
of Boltzmann entropy as a measure of information yielded by an
elementary statistical experiment and basic coding theorems on
storing messages and transmitting them through noisy communication
channels in an optimal manner; the influence of ergodic theory in
the enlargement of the scope of Shannon's theorems through the
works of McMillan, Feinstein, Wolfowitz, Breiman and others and its
impact on the appearance of the Kolmogorov-Sinai invariant for
elementary dynamical systems; and finally, the more recent work of
Schumacher, Holevo, Winter and others on the role of von Neumann
entropy in the quantum avatar of the basic coding theorems when
messages are encoded as quantum states, transmitted through noisy
quantum channels and retrieved by generalized measurements.
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