Potential Theory and Right Processes (Mathematics and Its Applications)

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.

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)