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Synopsis
This book develops the potential theory starting from a sub-Markovian resolvent of kernels on a measurable space, covering the context offered by a right process with general state space. It turns out that the main results from the classical cases (e.g., on locally compact spaces, with Green functions) have meaningful extensions to this setting. The study of the strongly supermedian functions and specific methods like the Revuz correspondence, for the largest class of measures, and the weak duality between two sub-Markovian resolvents of kernels are presented for the first time in a complete form. It is shown that the quasi-regular semi-Dirichlet forms fit in the weak duality hypothesis. Further results are related to the subordination operators and measure perturbations. The subject matter is supplied with a probabilistic counterpart, involving the homogeneous random measures, multiplicative, left and co-natural additive functionals.
The book is almost self-contained, being accessible to graduate students.
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Review
From the reviews:
"This book contains various topics on the general theory related to the analytic treatments of sub-Markovian resolvents, it will be a good reference for the specialists of the field. ... In each chapter, after the analytic arguments of the topics of the chapter, related probabilistic results are stated." (Yoichi Oshima, Zentralblatt MATH, Vol. 1091 (17), 2006)
"In the book under review, starting from a given sub-Markovian resolvent kernel {Ua} on a Radon measure space E, the authors consider analytic counterparts of the probability topics in this general framework. The book contains various subjects on the general theory involving the analytic treatments of sub-Markovian resolvents; it will be a good reference for specialists in the field." (Yoichi Oshima, Mathematical Reviews, Issue 2007 a)
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- PublisherSpringer
- Publication date2010
- ISBN 10 9048166713
- ISBN 13 9789048166718
- BindingPaperback
- Number of pages376
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Potential Theory and Right Processes (Mathematics and Its Applications) by Beznea, Lucian, Boboc, Nicu [Paperback ]
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Potential Theory and Right Processes
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ISBN 10: 9048166713
ISBN 13: 9789048166718
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Taschenbuch. Condition: Neu. This item is printed on demand - it takes 3-4 days longer - Neuware -The developmentsin the recent yearsof the potential theoryemphasized a classof functions larger than that of excessive functions (i.e. the positive superharmonic functionsfromtheclassicalpotentialtheoryassociatedwiththeLaplaceopera tor), namely the strongly supermedian functions. It turns out that a positive Borel function will be strongly supermedian if and only if it is the in mum of all its excessive majorants. Apparently, these functions have been introduced by J.F. Mertens and then they have been studied mainly by P.A. Meyer, G. Mokobodzki, D. Feyel and recently by P.J. Fitzsimmons and R.K. Getoor. The aimofthis bookisamongothersto developa potential theoryappropriate to this new class of functions. Although our methods are analytical, we present also the probabilistic counterparts from the Markov processes theory. The natural frame in which this theory is settled is given by a sub-Markovian resolvent of kernels on a Radon measurable space. After a possible extension of the space, such a resolvent becomes that one associated with a right process on a Radon topological space, not necessary locally compact and without existing a reference measure. Intimately related to the excessive functions we present certain basic tools of the theory: the Ray topology and compacti cation, the ne carrier and the reduction operation on measurable sets. We examine di erent types of negligible sets with respect to a nite measure :the -polar, -semipolar and -mince sets. We take advantage of the cone of potentials structure for both excessive functions and measures. 380 pp. Englisch. Seller Inventory # 9789048166718
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Potential Theory and Right Processes
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Potential Theory and Right Processes
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Taschenbuch. Condition: Neu. Druck auf Anfrage Neuware - Printed after ordering - The developmentsin the recent yearsof the potential theoryemphasized a classof functions larger than that of excessive functions (i.e. the positive superharmonic functionsfromtheclassicalpotentialtheoryassociatedwiththeLaplaceoperator), namely the strongly supermedian functions. It turns out that a positive Borel function will be strongly supermedian if and only if it is the in mum of all its excessive majorants. Apparently, these functions have been introduced by J.F. Mertens and then they have been studied mainly by P.A. Meyer, G. Mokobodzki, D. Feyel and recently by P.J. Fitzsimmons and R.K. Getoor. The aimofthis bookisamongothersto developa potential theoryappropriate to this new class of functions. Although our methods are analytical, we present also the probabilistic counterparts from the Markov processes theory. The natural frame in which this theory is settled is given by a sub-Markovian resolvent of kernels on a Radon measurable space. After a possible extension of the space, such a resolvent becomes that one associated with a right process on a Radon topological space, not necessary locally compact and without existing a reference measure. Intimately related to the excessive functions we present certain basic tools of the theory: the Ray topology and compacti cation, the ne carrier and the reduction operation on measurable sets. We examine di erent types of negligible sets with respect to a nite measure :the -polar, -semipolar and -mince sets. We take advantage of the cone of potentials structure for both excessive functions and measures. Seller Inventory # 9789048166718
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Condition: New. Dieser Artikel ist ein Print on Demand Artikel und wird nach Ihrer Bestellung fuer Sie gedruckt. Introduction. 1: Excessive Functions. 1.1. Sub-Markovian resolvent of kernels. 1.2. Basics on excessive functions. 1.3. Fine topology. 1.4. Excessive measures. 1.5. Ray topology and compactification. 1.6. The reduction operation and the associated capacitie. Seller Inventory # 5820521
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