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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.

A random cumulative distribution function (cdf) $F$ on $\\lbrack 0, \\infty)$ from a beta-Stacy process is defined. It is shown to be neutral to the right and a generalization of the Dirichlet process. The posterior distribution is also a beta-Stacy process given independent and identically distributed (iid) observations, possibly with right censoring, from $F$. A generalization of the Polya-urn scheme is introduced which characterizes the discrete beta-Stacy process.

Given the one-to-one correspondence between nearly Borel right processes and non-symmetric Dirichlet forms, we prove in the present paper that the killing transform of Markov processes is equivalent to strong subordination of the respective Dirichlet forms and give a characterization of so-called bivariate smooth measures.

Following Choquet, the capacity associated with a Markov process is said to be dichotomous if each compact set K contains two disjoint sets with the same capacity as K. In the context of right processes, we prove that the dichotomy of capacity is equivalent to Hunt's hypothesis that semipolar sets are polar. We also show that a weaker form of the dichotomy is valid for any Levy process with absolutely continuous potential kernel.

We show that a continuous-time Markov process X is Harris recurrent if and only if there exists a nonzero -finite measure on its state space such that X surely hits sets with positive -measure. This simple criterion is applied to some nonparametric closed queueing networks.

Using the continuum hypothesis, we produce an example answering three problems from the theory of right processes. In particular, we give a nontrivial example of a strong Markov process which is not a right process.

We associate with a strong Markov process $(X_t)$ and a Borel set $B$ an \"exit system.\" This system provides the structure of the excursions from $B$ of the process $(X_t)$ and gives a new approach to the recent results of Getoor and Sharpe on last exit decompositions and last exit distributions.

Let {Xt(ω)}\\{ {X_t}(\\omega )\\} represent a version of the Wiener process having almost surely continuous sample paths on (−∞,∞)( - \\infty ,\\infty ) that vanish at zero. We present a theorem concerning the local nature of the sample paths. Almost surely the local behavior at each t is of one of seven varieties thus inducing a partition of (−∞,∞)( - \\infty ,\\infty ) into seven disjoint Borel sets of the second class. The process {Xt(ω)}\\{ {X_t}(\\omega )\\} can be modified so that almost surely the sample paths are everywhere locally recurrent.