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Markov processes represent a universal model for a large variety of
real life random evolutions. The wide flow of new ideas, tools,
methods and applications constantly pours into the ever-growing
stream of research on Markov processes that rapidly spreads over
new fields of natural and social sciences, creating new streamlined
logical paths to its turbulent boundary. Even if a given process is
not Markov, it can be often inserted into a larger Markov one
(Markovianization procedure) by including the key historic
parameters into the state space. This monograph gives a concise,
but systematic and self-contained, exposition of the essentials of
Markov processes, together with recent achievements, working from
the "physical picture" - a formal pre-generator, and stressing the
interplay between probabilistic (stochastic differential equations)
and analytic (semigroups) tools. The book will be useful to
students and researchers. Part I can be used for a one-semester
course on Brownian motion, Levy and Markov processes, or on
probabilistic methods for PDE. Part II mainly contains the author's
research on Markov processes. From the contents: Tools from
Probability and Analysis Brownian motion Markov processes and
martingales SDE, DE and martingale problems Processes in Euclidean
spaces Processes in domains with a boundary Heat kernels for
stable-like processes Continuous-time random walks and fractional
dynamics Complex chains and Feynman integral
The first chapter deals with idempotent analysis per se . To make
the pres- tation self-contained, in the first two sections we
define idempotent semirings, give a concise exposition of
idempotent linear algebra, and survey some of its applications.
Idempotent linear algebra studies the properties of the semirn-
ules An , n E N , over a semiring A with idempotent addition; in
other words, it studies systems of equations that are linear in an
idempotent semiring. Pr- ably the first interesting and nontrivial
idempotent semiring , namely, that of all languages over a finite
alphabet, as well as linear equations in this sern- ing, was
examined by S. Kleene [107] in 1956 . This noncommutative semiring
was used in applications to compiling and parsing (see also [1]) .
Presently, the literature on idempotent algebra and its
applications to theoretical computer science (linguistic problems,
finite automata, discrete event systems, and Petri nets),
biomathematics, logic , mathematical physics , mathematical
economics, and optimizat ion, is immense; e. g. , see [9, 10, 11,
12, 13, 15, 16 , 17, 22, 31 , 32, 35,36,37,38,39 ,40,41,52,53
,54,55,61,62 ,63,64,68, 71, 72, 73,74,77,78,
79,80,81,82,83,84,85,86,88,114,125 ,128,135,136,
138,139,141,159,160,
167,170,173,174,175,176,177,178,179,180,185,186 , 187, 188, 189].
In 1. 2 we present the most important facts of the idempotent
algebra formalism . The semimodules An are idempotent analogs of
the finite-dimensional v- n, tor spaces lR and hence endomorphisms
of these semi modules can naturally be called (idempotent) linear
operators on An .
The monograph is devoted mainly to the analytical study of the differential, pseudo-differential and stochastic evolution equations describing the transition probabilities of various Markov processes. These include (i) diffusions (in particular,degenerate diffusions), (ii) more general jump-diffusions, especially stable jump-diffusions driven by stable Lévy processes, (iii) complex stochastic Schrödinger equations which correspond to models of quantum open systems. The main results of the book concern the existence, two-sided estimates, path integral representation, and small time and semiclassical asymptotics for the Green functions (or fundamental solutions) of these equations, which represent the transition probability densities of the corresponding random process. The boundary value problem for Hamiltonian systems and some spectral asymptotics ar also discussed. Readers should have an elementary knowledge of probability, complex and functional analysis, and calculus.
The first chapter deals with idempotent analysis per se . To make
the pres- tation self-contained, in the first two sections we
define idempotent semirings, give a concise exposition of
idempotent linear algebra, and survey some of its applications.
Idempotent linear algebra studies the properties of the semirn-
ules An , n E N , over a semiring A with idempotent addition; in
other words, it studies systems of equations that are linear in an
idempotent semiring. Pr- ably the first interesting and nontrivial
idempotent semiring , namely, that of all languages over a finite
alphabet, as well as linear equations in this sern- ing, was
examined by S. Kleene [107] in 1956 . This noncommutative semiring
was used in applications to compiling and parsing (see also [1]) .
Presently, the literature on idempotent algebra and its
applications to theoretical computer science (linguistic problems,
finite automata, discrete event systems, and Petri nets),
biomathematics, logic , mathematical physics , mathematical
economics, and optimizat ion, is immense; e. g. , see [9, 10, 11,
12, 13, 15, 16 , 17, 22, 31 , 32, 35,36,37,38,39 ,40,41,52,53
,54,55,61,62 ,63,64,68, 71, 72, 73,74,77,78,
79,80,81,82,83,84,85,86,88,114,125 ,128,135,136,
138,139,141,159,160,
167,170,173,174,175,176,177,178,179,180,185,186 , 187, 188, 189].
In 1. 2 we present the most important facts of the idempotent
algebra formalism . The semimodules An are idempotent analogs of
the finite-dimensional v- n, tor spaces lR and hence endomorphisms
of these semi modules can naturally be called (idempotent) linear
operators on An .
There has been an increase in attention toward systems involving
large numbers of small players, giving rise to the theory of mean
field games, mean field type control and nonlinear Markov games.
Exhibiting various real world problems involving major and minor
agents, this book presents a systematic continuous-space
approximation approach for mean-field interacting agents models and
mean-field games models. After describing Markov-chain methodology
and a modeling of mean-field interacting systems, the text presents
various structural conditions on the chain to yield respective
socio-economic models, focusing on migration models via binary
interactions. The specific applications are wide-ranging -
including inspection and corruption, cyber-security,
counterterrorism, coalition building and network growth, minority
games, and investment policies and optimal allocation - making this
book relevant to a wide audience of applied mathematicians
interested in operations research, computer science, national
security, economics, and finance.
There has been an increase in attention toward systems involving
large numbers of small players, giving rise to the theory of mean
field games, mean field type control and nonlinear Markov games.
Exhibiting various real world problems involving major and minor
agents, this book presents a systematic continuous-space
approximation approach for mean-field interacting agents models and
mean-field games models. After describing Markov-chain methodology
and a modeling of mean-field interacting systems, the text presents
various structural conditions on the chain to yield respective
socio-economic models, focusing on migration models via binary
interactions. The specific applications are wide-ranging -
including inspection and corruption, cyber-security,
counterterrorism, coalition building and network growth, minority
games, and investment policies and optimal allocation - making this
book relevant to a wide audience of applied mathematicians
interested in operations research, computer science, national
security, economics, and finance.
A nonlinear Markov evolution is a dynamical system generated by a
measure-valued ordinary differential equation with the specific
feature of preserving positivity. This feature distinguishes it
from general vector-valued differential equations and yields a
natural link with probability, both in interpreting results and in
the tools of analysis. This brilliant book, the first devoted to
the area, develops this interplay between probability and analysis.
After systematically presenting both analytic and probabilistic
techniques, the author uses probability to obtain deeper insight
into nonlinear dynamics, and analysis to tackle difficult problems
in the description of random and chaotic behavior. The book
addresses the most fundamental questions in the theory of nonlinear
Markov processes: existence, uniqueness, constructions,
approximation schemes, regularity, law of large numbers and
probabilistic interpretations. Its careful exposition makes the
book accessible to researchers and graduate students in stochastic
and functional analysis with applications to mathematical physics
and systems biology.
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