Potential Theory and Right Processes
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Author Description
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.
Menrva's Summary
THE THESIS
This book investigates the fundamental principles of potential theory by extending classical results to the context of right processes with general state spaces. The author, Lucian Beznea, builds upon the foundation of sub-Markovian resolvents of kernels to explore advanced concepts such as strongly supermedian functions, the Revuz correspondence, and weak duality between resolvents. The work aims to provide a comprehensive treatment of these topics, making them accessible to graduate students while also offering novel insights into measure perturbations and subordination operators.
THE SCOPE MAP
- Development of potential theory starting from a sub-Markovian resolvent of kernels.
- Coverage of right processes with general state spaces.
- Extensions of classical results (e.g., on locally compact spaces, with Green functions).
- Detailed presentation of strongly supermedian functions.
- Inclusion of specific methods like the Revuz correspondence for the largest class of measures.
- First-time complete presentation of weak duality between two sub-Markovian resolvents of kernels.
- Demonstration of how quasi-regular semi-Dirichlet forms fit into the weak duality hypothesis.
- Exploration of subordination operators and measure perturbations.
- Probabilistic counterpart involving homogeneous random measures, multiplicative, left, and co-natural additive functionals.
THE AUTHORITY PERSPECTIVE
The book is presented as a comprehensive and accessible graduate-level text on potential theory, extending classical concepts to more general settings. The inclusion of topics like the Revuz correspondence and weak duality in their complete form, along with their probabilistic counterparts, suggests a rigorous and thorough treatment. The author's aim to make the material self-contained indicates a focus on pedagogical clarity for students entering the field. The detailed scope suggests it will be a valuable reference for researchers in stochastic processes and related areas.
Page Count:
380
Publication Date:
2010-01-01
ISBN-10:
9048166713
ISBN-13:
9789048166718
Science
Probability Theory
Stochastic processes
Mathematics
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