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.
Table of ContentsIntroduction.
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 capacities. 1.7. Polar and semipolar sets. Nearly measurable functions. 1.8. Probabilistic interpretations: Sub-Markovian resolvents and right processes.
2: Cones of Potentials and H Cones. 2.1. Basics on cones of potentials and H-cones. 2.2. sigma-Balayages on cones of potentials. 2.3. Balayages on H-cones. 2.4. Quasi bounded, subtractive and regular elements of a cone of potentials.
3: Fine Potential Theoretical Techniques. 3.1. Cones of potentials associated with a sub-Markovian resolvent. 3.2. Regular excessive functions, fine carrier and semipolarity. 3.3. Representation of balayages on excessive measures. 3.4. Quasi bounded, regular and subtractive excessive measures. 3.5. Tightness for sub-Markovian resolvents. 3.6. Localization in excessive functions and excessive measures. 3.7. Probabilistic interpretations: Continuous additive functionals and standardness.
4: Strongly Supermedian Functions and Kernels. 4.1. Supermedian functionals. 4.2. Supermedian lambda-quasi kernels. 4.3. Strongly supermedian functions. 4.4. Fine densities. 4.5. Probabilistic interpretations: Homogeneous random measures.
5: Subordinate Resolvents. 5.1. Weak subordination operators. 5.2. Inverse subordination. 5.3. Probabilistic interpretations: Multiplicative functionals.
6: Revuz Correspondence. 6.1. Revuz measures. 6.2. Hypothesis (i) of Hunt. 6.3. Smooth measures and sub-Markovian resolvents. 6.4. Measure perturbation of sub-Markovian resolvents. 6.5. Probabilistic interpretations: Positive left additive functionals.
7: Resolvents under Weak Duality Hypothesis. 7.1. Weak duality hypothesis. 7.2. Natural potential kernels and the Revuz correspondence. 7.3. Smooth and cosmooth measures. 7.4. Subordinate resolvents in weak duality. 7.5. Semi-Dirichlet forms. 7.6. Weak duality induced by a semi-Dirichlet form. 7.7. Probabilistic interpretations: Multiplicative functionals in weak duality.
A. Appendix: A.1. Complements on measure theory, kernels, Choquet boundary and capacity. A.2. Complements on right processes. A.3. Cones of potentials and H-cones. A.4. Basics on coercive closed bilinear forms.
Notes. Bibliography. Index.