In this volume two topics are discussed:
the construction of
Feller and Lp-sub-Markovian semigroups by
starting with a pseudo-differential
operator, and the potential theory of these
semigroups and their
generators. The first part of the text essentially
discusses the
analysis of pseudo-differential operators
with negative definite
symbols and develops a symbolic calculus;
in addition, it deals
with special approaches, such as subordination
in the sense of
Bochner. The second part handles capacities,
function spaces
associated with continuous negative definite
functions, Lp-sub-Markovian
semigroups in their associated Bessel potential
spaces, Stein's
Littlewood?Paley theory, global properties
of Lp?sub-Markovian
semigroups, and estimates for transition
functions.
Contents:
Generators of Feller and Sub-Markovian Semigroups:
Second Order Elliptic Differential Operators as Generators of Feller and Sub-Markovian Semigroups
Some Second Order Differential Operators with Non-Negative Characteristic Form as Generators of Sub-Markovian Semigroups
Some Properties of Pseudo-Differential Operators with Negative Definite Symbols
Hoh's Symbolic Calculus for Pseudo-Differential Operators with Nagative Definite Symbols
Estimates for Pseudo-Differential Operators with Negative Definite Symbols Using the Symbolic Calculus
Feller Semigroups and Sub-Markovian Semigroups Generated by Pseudo-Differential Operators
Further Analytic Approaches for Constructing Feller and Sub-Markovian Semigroups
Some Perturbation Results
On Semigroups Obtained by Subordination
Pseudo-Differential Operators with Variable Order of Differentiation as Generators of Feller Semigroups
Potential Theory of Semigroups and Generators:
Capacities and Abstract Bessel Potential Spaces
First Results on Lp-Sub-Markovian Semigroups in Their Associated Bessel Potential Spaces
Bessel Potential Spaces Assocated with a Continuous Negative Definite Function
Stein's Littlewood?Paley Theory for Sub-Markovian Semigroups
Global Properties of Lp-Sub-Markovian Semigroups
Nash-Type and Sobolev-Type Inequalities ? A Short Outline
Readership: Graduate students, lecturers
and researchers in the fields of analysis
& differential equations, stochastic,
probability & statistics, and mathematical
physics.
500pp Pub. date: Scheduled Fall 2002
ISBN 1-86094-324-1
This book introduces an analytically tractable
and
computationally effective class of non-Gaussian
models for shocks
(regular Levy processes of the exponential
type) and related
analytical methods similar to the initial
Merton?Black?Scholes
approach, which the authors call the Merton?Black?Scholes
theory.
The authors have chosen applications interesting
for financial
engineers and specialists in financial economics,
real options,
and partial differential equations (especially
pseudodifferential
operators); specialists in stochastic processes
will benefit from
the use of the pseudodifferential operators
technique in non-Gaussian
situations. The authors also consider discrete
time analogues of
perpetual American options and the problem
of the optimal choice
of capital, and outline several possible
directions in which the
methods of the book can be developed further.
Taking account of a diverse audience, the
book has been written
in such a way that it is simple at the beginning
and more
technical in further chapters, so that it
is accessible to
graduate students in relevant areas and mathematicians
without
prior knowledge of finance or economics.
Contents:
Levy Processes
Regular Levy Processes of Exponential Type in 1D
Pricing and Hedging of Contingent Claims of European Type
Perpetual American Options
American Options: Finite Time Horizon
First-Touch Digitals
Barrier Options
Multi-Asset Contracts
Investment Under Uncertainty and Capital Accumulation
Endogenous Default and Pricing of the Corporate Debt
Fast Pricing of European Options
Discrete Time Models
Feller Processes of Normal Inverse Gaussian Type
Pseudodifferential Operators with Constant Symbols
Elements of Calculus of Pseudodifferential Operators
Readership: Graduate students, researchers
and academics in economics, mathematical
finance, banking & finance/accounting;
and financial engineers.
420pp Pub. date: Apr 2002
ISBN 981-02-4944-6
This book is dedicated to the memory of Michael
Marinov, the
theorist who, together with Felix Berezin,
introduced the
classical description of spin by anticommuting
Grassmann
variables. It contains original papers and
reviews by physicists
and mathematicians written specifically for
the book. These
articles reflect the current status and recent
developments in
the areas of Marinov's research: quantum
tunneling, quantization
of constrained systems, supersymmetry, and
others. The personal
recollections included portray the human
face of M Marinov, a
person of great knowledge and integrity.
Contents:
Personal Notes:
My Brother (I Kwartin)
My Dear Old Friend Misha (L Brink)
Quantum Tunneling, Coherence and Decoherence:
Quantum Tunneling (M Shifman)
Evolution Kernels for Phase-Space Distributions (B Segev)
Quantization of Constrained System, Integrable Models:
Lie Algebroids as Gauge Symmetries in Topological Field Theories (M Olshanetsky)
Classical Integrable System: Selected Topics (A Perelomov)
Quantum Chromodynamics, Hadrons and Fields:
Probabilistic Model of Reggeon Field Theory (K G Boreskov)
Diffractive Processes and QCD (A B Kaidalov)
Mathematical and Physical Aspects of Supersymmetry:
A Theory of Algebraic Integration (R Casalbuoni)
The Witten Index Beyond the Adiabatic Approximation (P van Baal)
Other Aspects:
Cube or Hypercube of Natural Units (L B Okun)
Approximation Methods in Classical and Quantum Theories (R Kerner)
Strings, Branes and Higher Dimensions:
Aspects of Tachyon Condensation in String Theory (Y Oz)
String Tension in Two-Dimensional Gauge Theories (A Armoni et al.)
and other articles
Readership: Graduate students, academics
and researchers in theoretical, mathematical
and high energy physics.
900pp (approx.) Pub. date: Scheduled Fall 2002
ISBN 981-238-072-8
The dynamics of infinite classical lattice
systems has been
considered and has led to the study of the
properties of
ergodicity and convergence to equilibrium
of a new class of
Markov semigroups. Quantum analogues of these
semigroups have
also been considered. However, the problem
of deriving these
Markovian semigroups and, what is much more
interesting, the
associated stochastic flows, as limits of
Hamiltonian systems,
rather than postulating their form on a phenomenological
basis,
is essentially open both in the classical
case and in the quantum
case.
This book presents a conjecture that, by
coupling a quantum spin
system in finite volume to a quantum field
via a suitable
interaction, applying the stochastic golden
rule and taking the
thermodynamic limit, one may obtain a class
of quantum flows
which, when restricted to an appropriate
Abelian subalgebra,
gives rise to the classical interacting particle
systems studied
in classical statistical mechanics.
In the first chapter of the book, it is proved
that this
conjecture is true and that the class of
quantum-dynamical
semigroups arising from the stochastic limit,
in the weak
coupling regime, has a rich structure which
allows one in some
cases to write down explicitly their invariant
or equilibrium
distributions. Chapter 2 discusses simple
and effective methods
to analyze qualitatively the behavior of
quantum Markov
semigroups. The general methods discussed
in the first two
chapters are mainly effective in finite volume.
New ideas and
techniques which are specific to certain
classes of generators
need to be developed for use in infinite
volume; this is the
subject of the last chapter.
Contents:
Lectures on Quantum Interacting Particle
Systems (L Accardi & S Kozyrev):
Basic Ideas of the Stochastic Limit
Equilibrium and Nonequilibrium Spin-Boson
Models
Dynamics of Quantum Spin Systems
Structure of the Markov Flows
Existence of Infinite Volume Flows for Quantum
Spin Systems
Appendix: Basic Notions on Semigroups and
Dissipations
Lectures on the Qualitative Analysis of Quantum
Markov Semigroups
(F Fagnola & R Reborello):
Quantum Markov Semigroups and Master Equations
The Existence of Stationary States
Faithful Stationary States and Irreducibility
The Convergence Towards the Equilibrium
Analysis of Classical and Quantum Interacting
Particle Systems (B
Zegarlinski):
Construction and Ergodicity of Jump Type
Dynamics
Hypercontractivity and Strong Ergodicity
for Classical Systems
Logarithmic Sobolev Inequality in Non-Commutative
Setting
Particle Structure and Hypercontractivity
Readership: Researchers in probability &
statistics, stochastic, mathematical physics,
quantum physics.
350pp (approx.) Pub. date: Scheduled Fall 2002
ISBN 981-238-104-X
This important book explains how the technique of Witten
Laplacians may be useful in statistical mechanics. It considers
the problem of analyzing the decay of correlations, after
presenting its origin in statistical mechanics. In addition, it
compares the Witten Laplacian approach with other techniques,
such as the transfer matrix approach and its semiclassical
analysis. The author concludes by providing a complete proof of
the uniform Log?Sobolev inequality.
Readership: Graduate students and researchers in the fields of
partial differential equations, mathematical physics and
statistical physics.
180pp (approx.) Pub. date: Scheduled Fall 2002
ISBN 981-238-098-1