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, the book covers 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 book develops 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. It then addresses 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 book 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.
In this paper, we investigate symmetric jump-type processes on a class of metric measure spaces with jumping intensities comparable to radially symmetric functions on the spaces. The class of metric measure spaces includes the Alfors d-regular sets, which is a class of fractal sets that contains geometrically self-similar sets. A typical example of our jump-type processes is the symmetric jump process with jumping intensitywhere ν is a probability measure on [α 1 , α 2 ] ⊂ (0, 2), c(α, x, y) is a jointly measurable function that is symmetric in (x, y) and is bounded between two positive constants, and c 0 (x, y) is a jointly measurable function that is symmetric in (x, y) and is bounded between γ 1 and γ 2 , where either γ 2 ≥ γ 1 > 0 or γ 1 = γ 2 = 0. This example contains mixed symmetric stable processes on R n as well as mixed relativistic symmetric stable Dedicated to Professor Masatoshi Fukushima on the occasion of his 70th birthday.
Abstract. We consider the non-local symmetric Dirichlet form (E, F ) given bywith F the closure with respect to E 1 of the set of C 1 functions on R d with compact support, where, and where the jump kernel J satisfiesfor 0 < α < β < 2, |x − y| < 1. This assumption allows the corresponding jump process to have jump intensities whose sizes depend on the position of the process and the direction of the jump. We prove upper and lower estimates on the heat kernel. We construct a strong Markov process corresponding to (E, F ). We prove a parabolic Harnack inequality for non-negative functions that solve the heat equation with respect to E. Finally we construct an example where the corresponding harmonic functions need not be continuous.
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