Abstract
In the next two sections our aim is to establish a number of notable results in classical potential theory by the methods developed in the earlier chapters. In contrast with the preceding sections of last chapter, the horizon will be widened to reach far beyond Brownian motion. We shall deal with Hunt processes satisfying certain general hypotheses and the results will apply to classes of potential kernels including the M. Riesz potentials (see Exercises below) as well as the logarithmic and Newtonian. There are usually different sets of overlapping conditions to yield a particular result. The theory of dual processes in Blumenthal and Getoor [1] gives a framework which has a considerable range, but it is a long and sometimes technically complicated passage. Here instead we offer a more direct and relatively new approach to a number of selected topics, with a view to further development. The case of Brownian motion will be discussed toward the end.
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Notes on Chapter 5
The title of this chapter is a double entendre. It is not the intention of this book to treat potential theory except as a concomitant of the underlying processes. Nevertheless we shall proceed far enough to show several facets of this old theory in a new light.
The content of this section is based on Chung [1]. This was an attempt to utilize the inherent duality of a Markov process as evidenced by its hitting and quitting times. Formula (20) or its better known corollary (21) is a particular case of the representation of an excessive function by the potential of a measure plus a generalized harmonic function. Such a representation is proved in Chung and Rao [1] for the class of potentials which satisfies the conditions of Theorem 1. In classical potential theory this is known as F. Riesz’s decomposition of a subharmonic function; see Brelot [1]. We stop short of this fundamental result as it would take us too far into a well-entrenched field.
Let us point out that Robin’s problem of determining the equilibrium charge distribution is solved by the formula (14). One might employ a kind of Monte Carlo method to simulate the last exit probability there to obtain empirical results.
The various principles of Newtonian potential theory are discussed in Rao [1]. The extension of these to M. Riesz potentials was a major advance made by Frostman [1]. Their mutual relations form the base of an axiomatic potential theory founded by Brelot. Wermer [1] gives a brief account of the physical concepts of capacity and energy leading to the equilibrium potential.
In principle, it should be possible to treat questions of duality by the method of time reversal. Although a general theory of reversing has existed for some time, it seems still too difficult for applications. A prime example is the polarity principle (Theorem 3) which is not prima facie such a problem. Yet all known probabilistic proofs use some kind of reversing (cf. Theorem 10 of §4.5). It would be extremely interesting to obtain this and perhaps also the maximum principle (Theorem 2) by a manifest reversing argument. Such an approach is also suggested by the concept of reversibility of physical processes from which potential theory sprang.
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© 1982 Springer Science+Business Media New York
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Chung, K.L. (1982). Potential Developments. In: Lectures from Markov Processes to Brownian Motion. Grundlehren der mathematischen Wissenschaften, vol 249. Springer, New York, NY. https://doi.org/10.1007/978-1-4757-1776-1_5
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DOI: https://doi.org/10.1007/978-1-4757-1776-1_5
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