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The book is devoted to the mathematical foundations of nonextensive
statistical mechanics. This is the first book containing the
systematic presentation of the mathematical theory and concepts
related to nonextensive statistical mechanics, a current
generalization of Boltzmann-Gibbs statistical mechanics introduced
in 1988 by one of the authors and based on a nonadditive entropic
functional extending the usual Boltzmann-Gibbs-von Neumann-Shannon
entropy. Main mathematical tools like the q-exponential function,
q-Gaussian distribution, q-Fourier transform, q-central limit
theorems, and other related objects are discussed rigorously with
detailed mathematical rational. The book also contains recent
results obtained in this direction and challenging open problems.
Each chapter is accompanied with additional useful notes including
the history of development and related bibliographies for further
reading.
The book is devoted to the fundamental relationship between three
objects: a stochastic process, stochastic differential equations
driven by that process and their associated
Fokker-Planck-Kolmogorov equations. This book discusses wide
fractional generalizations of this fundamental triple relationship,
where the driving process represents a time-changed stochastic
process; the Fokker-Planck-Kolmogorov equation involves
time-fractional order derivatives and spatial pseudo-differential
operators; and the associated stochastic differential equation
describes the stochastic behavior of the solution process. It
contains recent results obtained in this direction.This book is
important since the latest developments in the field, including the
role of driving processes and their scaling limits, the forms of
corresponding stochastic differential equations, and associated FPK
equations, are systematically presented. Examples and important
applications to various scientific, engineering, and economics
problems make the book attractive for all interested researchers,
educators, and graduate students.
The book systematically presents the theories of
pseudo-differential operators with symbols singular in dual
variables, fractional order derivatives, distributed and variable
order fractional derivatives, random walk approximants, and
applications of these theories to various initial and multi-point
boundary value problems for pseudo-differential equations.
Fractional Fokker-Planck-Kolmogorov equations associated with a
large class of stochastic processes are presented. A complex
version of the theory of pseudo-differential operators with
meromorphic symbols based on the recently introduced complex
Fourier transform is developed and applied for initial and boundary
value problems for systems of complex differential and
pseudo-differential equations.
The book systematically presents the theories of
pseudo-differential operators with symbols singular in dual
variables, fractional order derivatives, distributed and variable
order fractional derivatives, random walk approximants, and
applications of these theories to various initial and multi-point
boundary value problems for pseudo-differential equations.
Fractional Fokker-Planck-Kolmogorov equations associated with a
large class of stochastic processes are presented. A complex
version of the theory of pseudo-differential operators with
meromorphic symbols based on the recently introduced complex
Fourier transform is developed and applied for initial and boundary
value problems for systems of complex differential and
pseudo-differential equations.
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