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Stochastic processes --- Functional analysis --- 519.2 <061.3> --- Probability. Mathematical statistics--?<061.3> --- Markov processes --- Congresses. --- 519.2 <061.3> Probability. Mathematical statistics--?<061.3> --- Analyse fonctionnelle --- Functional analysis. --- Markov, Processus de

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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.

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This book contains an introductory and comprehensive account of the theory of (symmetric) Dirichlet forms. Moreover this analytic theory is unified with the probabilistic potential theory based on symmetric Markov processes and developed further in conjunction with the stochastic analysis based on additive functional. Since the publication of the first edition in 1994, this book has attracted constant interests from readers and is by now regarded as a standard reference for the theory of Dirichlet forms. For the present second edition, the authors not only revised the existing text, but also added sections on capacities and Sobolev type inequalities, irreducible recurrence and ergodicity, recurrence and Poincaré type inequalities, the Donsker-Varadhan type large deviation principle, as well as several new exercises with solutions. The book addresses to researchers and graduate students who wish to comprehend the area of Dirichlet forms and symmetric Markov processes.

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