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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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Dirichlet forms. --- Forms, Dirichlet --- Forms (Mathematics) --- Formes (Matemàtica) --- Probabilitats --- Càlcul de probabilitats --- Inferència estadística --- Probabilitat --- Combinatòria (Matemàtica) --- Lògica --- Lògica matemàtica --- Anàlisi de sèries temporals --- Correlació (Estadística) --- Descomposició (Matemàtica) --- Distribució (Teoria de la probabilitat) --- Fiabilitat (Enginyeria) --- Funcions característiques --- Geometria estocàstica --- Incertesa (Teoria de la informació) --- Integrals de camí --- Jocs d'atzar (Matemàtica) --- Mitjana (Estadística) --- Probabilitats combinatòries --- Processos estocàstics --- Sort --- Teoremes de límit (Teoria de probabilitats) --- Teoria matemàtica de la comunicació --- Variables aleatòries --- Atzar --- Risc (Economia) --- Teoria ergòdica --- Àlgebra --- Matemàtica --- Formes automorfes --- Formes bilineals --- Formes modulars --- Polígons

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"In this paper, we consider symmetric jump processes of mixed-type on metric measure spaces under general volume doubling condition, and establish stability of two-sided heat kernel estimates and heat kernel upper bounds. We obtain their stable equivalent characterizations in terms of the jumping kernels, variants of cut-off Sobolev inequalities, and the Faber-Krahn inequalities. In particular, we establish stability of heat kernel estimates for -stable-like processes even with 2 when the underlying spaces have walk dimensions larger than 2, which has been one of the major open problems in this area"--

Kernel functions. --- Probability theory and stochastic processes -- Markov processes -- Transition functions, generators and resolvents. --- Partial differential equations -- Parabolic equations and systems -- Heat kernel. --- Probability theory and stochastic processes -- Markov processes -- Jump processes. --- Potential theory -- Other generalizations -- Dirichlet spaces. --- Probability theory and stochastic processes -- Markov processes -- Continuous-time Markov processes on general state spaces. --- Probability theory and stochastic processes -- Markov processes -- Probabilistic potential theory.

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This book contains original research papers by leading experts in the fields of probability theory, stochastic analysis, potential theory and mathematical physics. There is also a historical account on Masatoshi Fukushima's contribution to mathematics, as well as authoritative surveys on the state of the art in the field.

Mathematical analysis. --- Stochastic analysis. --- Probabilities. --- 517.1 Mathematical analysis --- Mathematical analysis --- Probability --- Statistical inference --- Combinations --- Mathematics --- Chance --- Least squares --- Mathematical statistics --- Risk --- Analysis, Stochastic --- Stochastic processes

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