1 Classical and Parabolic Potential Theory.- I Introduction to the Mathematical Background of Classical Potential Theory.- 1. The Context of Green s Identity.- 2. Function Averages.- 3. Harmonic Functions.- 4. Maximum-Minimum Theorem for Harmonic Functions.- 5. The Fundamental Kernel for ?N and Its Potentials.- 6. Gauss Integral Theorem.- 7. The Smoothness of Potentials; The Poisson Equation.- 8. Harmonic Measure and the Riesz Decomposition.- II Basic Properties of Harmonic, Subharmonic, and Superharmonic Functions.- 1. The Green Function of a Ball; The Poisson Integral.- 2. Harnack s Inequality.- 3. Convergence of Directed Sets of Harmonic Functions.- 4. Harmonic, Subharmonic, and Superharmonic Functions.- 5. Minimum Theorem for Superharmonic Functions.- 6. Application of the Operation ?B.- 7. Characterization of Superharmonic Functions in Terms of Harmonic Functions.- 8. Differentiate Superharmonic Functions.- 9. Application of Jensen s Inequality.- 10. Superharmonic Functions on an Annulus.- 11. Examples.- 12. The Kelvin Transformation (N ? 2).- 13. Greenian Sets.- 14. The L1(?B-) and D(?B-) Classes of Harmonic Functions on a Ball B; The Riesz-Herglotz Theorem.- 15. The Fatou Boundary Limit Theorem.- 16. Minimal Harmonic Functions.- III Infima of Families of Superharmonic Functions.- 1. Least Superharmonic Majorant (LM) and Greatest Subharmonic Minorant (GM).- 2. Generalization of Theorem 1.- 3. Fundamental Convergence Theorem (Preliminary Version).- 4. The Reduction Operation.- 5. Reduction Properties.- 6. A Smallness Property of Reductions on Compact Sets.- 7. The Natural (Pointwise) Order Decomposition for Positive Superharmonic Functions.- IV Potentials on Special Open Sets.- 1. Special Open Sets, and Potentials on Them.- 2. Examples.- 3. A Fundamental Smallness Property of Potentials.- 4. Increasing Sequences of Potentials.- 5. Smoothing of a Potential.- 6. Uniqueness of the Measure Determining a Potential.- 7. Riesz Measure Associated with a Superharmonic Function.- 8. Riesz Decomposition Theorem.- 9. Counterpart for Superharmonic Functions on ?2 of the Riesz Decomposition.- 10. An Approximation Theorem.- V Polar Sets and Their Applications.- 1. Definition.- 2. Superharmonic Functions Associated with a Polar Set.- 3. Countable Unions of Polar Sets.
Potential theory and certain aspects of probability theory are intimately related, perhaps most obviously in that the transition function determining a Markov process can be used to define the Green function of a potential theory. Thus it is possible to define and develop many potential theoretic concepts probabilistically, a procedure potential theorists observe withjaun diced eyes in view of the fact that now as in the past their subject provides the motivation for much of Markov process theory. However that may be it is clear that certain concepts in potential theory correspond closely to concepts in probability theory, specifically to concepts in martingale theory. For example, superharmonic functions correspond to supermartingales. More specifically: the Fatou type boundary limit theorems in potential theory correspond to supermartingale convergence theorems; the limit properties of monotone sequences of superharmonic functions correspond surprisingly closely to limit properties of monotone sequences of super martingales; certain positive superharmonic functions [supermartingales] are called "potentials," have associated measures in their respective theories and are subject to domination principles (inequalities) involving the supports of those measures; in each theory there is a reduction operation whose properties are the same in the two theories and these reductions induce sweeping (balayage) of the measures associated with potentials, and so on.
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