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This book deals with analytic treatments of Markov processes. Symmetric Dirichlet forms and their associated Markov processes are important and powerful tools in the theory of Markov processes and their applications. The theory is well studied and used in various fields. In this monograph, we intend to generalize the theory to non-symmetric and time dependent semi-Dirichlet forms. By this generalization, we can cover the wide class of Markov processes and analytic theory which do not possess the dual Markov processes. In particular, under the semi-Dirichlet form setting, the stochastic calculus is not well established yet. In this monograph, we intend to give an introduction to such calculus. Furthermore, basic examples different from the symmetric cases are given. The text is written for graduate students, but also researchers.

Markov processes. --- Dirichlet forms. --- Forms, Dirichlet --- Forms (Mathematics) --- Analysis, Markov --- Chains, Markov --- Markoff processes --- Markov analysis --- Markov chains --- Markov models --- Models, Markov --- Processes, Markov --- Stochastic processes --- Denumerable Structures. --- Diffusion Processes. --- Dirichlet Spaces. --- Probabilistic Potential Theory. --- Second-order Parabolic Equations.

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This book gives a comprehensive and self-contained introduction to the theory of symmetric Markov processes and symmetric quasi-regular Dirichlet forms. In a detailed and accessible manner, Zhen-Qing Chen and Masatoshi Fukushima cover the essential elements and applications of the theory of symmetric Markov processes, including recurrence/transience criteria, probabilistic potential theory, additive functional theory, and time change theory. The authors develop the theory in a general framework of symmetric quasi-regular Dirichlet forms in a unified manner with that of regular Dirichlet forms, emphasizing the role of extended Dirichlet spaces and the rich interplay between the probabilistic and analytic aspects of the theory. Chen and Fukushima then address the latest advances in the theory, presented here for the first time in any book. Topics include the characterization of time-changed Markov processes in terms of Douglas integrals and a systematic account of reflected Dirichlet spaces, and the important roles such advances play in the boundary theory of symmetric Markov processes. This volume is an ideal resource for researchers and practitioners, and can also serve as a textbook for advanced graduate students. It includes examples, appendixes, and exercises with solutions.

Markov processes. --- Boundary value problems. --- Dirichlet problem. --- Beurling-Deny decomposition. --- Beurling-Deny formula. --- Brownian motions. --- Dirichlet forms. --- Dirichlet spaces. --- Douglas integrals. --- Feller measures. --- Hausdorff topological space. --- Markovian symmetric operators. --- Silverstein extension. --- additive functional theory. --- additive functionals. --- analytic concepts. --- analytic potential theory. --- boundary theory. --- countable boundary. --- decompositions. --- energy functional. --- extended Dirichlet spaces. --- fine properties. --- harmonic functions. --- harmonicity. --- hitting distributions. --- irreducibility. --- lateral condition. --- local properties. --- m-tight special Borel. --- many-point extensions. --- one-point extensions. --- part processes. --- path behavior. --- perturbed Dirichlet forms. --- positive continuous additive functionals. --- probabilistic derivation. --- probabilistic potential theory. --- quasi properties. --- quasi-homeomorphism. --- quasi-regular Dirichlet forms. --- recurrence. --- reflected Dirichlet spaces. --- reflecting Brownian motions. --- reflecting extensions. --- regular Dirichlet forms. --- regular recurrent Dirichlet forms. --- smooth measures. --- symmetric Hunt processes. --- symmetric Markov processes. --- symmetric Markovian semigroups. --- terminal random variables. --- time change theory. --- time changes. --- time-changed process. --- transience. --- transient regular Dirichlet forms.