Abstract
In this chapter the state space E is the d-dimensional Euclidean space R d , d > 1 ; 𝓔 is the classical Borel field on R d . For A ∈𝓔, B∈ 𝓔, the sets A ± B are the vectorial sum and difference of A and B, namely the set of x ± y where x ∈ A, y ∈ B. When B = {x} this is written as A ± x; the set o — A is written as — A, where o is the zero point (origin) of R d.
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Notes on Chapter 4
The theory of spatially homogeneous Markov processes is an extension ofthat of random walks to the continuous parameter case. This is an old theory due largely to Paul Levy [1]. Owing to its special character classical methods of analysis such as the Fourier transform are applicable; see Gihman and Skorohod [1] for a more recent treatment.
For the theory of dual processes see Blumenthal and Getoor [1], which improved on Hunt’s original formulation. Much of the classical Newtonian theory is contained in the last few pages of the book in a condensed manner, but it is a remarkable synthesis not fully appreciated by the non-probabilists.
For lack of space we have to de-emphasize the case of dimension d = 1 or 2 in our treatment of Brownian motion. So far as feasible we use the general methods of Hunt processes and desist from unnecessary short-cuts. More coverage is available in the cognate books by K. M. Rao [1] and Port and Stone [1]. The former exposition takes a more general probabilistic approach while the latter has more details on several topics discussed here.
and §4.4. The force of the probabilistic method is amply illustrated in the solution of the Dirichlet problem. The reader who learns this natural approach first may indeed wonder at the tour de force of the classical treatments, in which some of the basic definitions such as the regularity of boundary would appear to be rather contrived.
As an introduction to the classical viewpoint the old book by Kellogg [1] is still valuable, particularly for its discussion of the physical background. A simpler version may be found in Wermer [1]. Ahlfors [1] contains an elementary discussion of harmonic functions and the Dirichlet problem in R 2, and the connections with analytic functions. Brelot [1] contains many modern developments as well as an elegant (French style) exposition of the Newtonian theory.
The proof of Theorem 8 by means of Lemma 9 may be new. The slow pace adopted here serves as an example of the caution needed in certain arguments. This is probably one of the reasons why even probabilists often bypass such proofs.
Another method of treating superharmonic functions is through approximation with smooth ones, based on results such as Theorem 12; see the books by Rao and Port-Stone. This approach leads to their deeper analysis as Schwartz distributions. We choose Doob’s method to give further credance to the viability of paths. This method is longer but ties several items together. The connections between (sub)harmonic functions and (sub)martingales were first explored in Doob [2], mainly for the logarithmic potential. In regard to Theorems 2 and 3, a detailed study of Brownian motion killed outside a domain requires the use of Green’s function, namely the density of the kernel Q t defined in (15), due to Hunt [1]. Here we regard the case as a worthy illustration of the general methodology B;{ and all).
Doob proved Theorem 9 in [2] using H. Cartan’s results on Newtionian capacity. A non-probabilistic proof of the Corollary to Theorem 10 can be found in Wermer [1]. The general proposition that “semipolar implies polar” is Hunt’s Hypothesis (H) and is one of the deepest results in potential theory. Several equivalent propositions are discussed in Blumen thai and Getoor [1]. A proof in a more general case than the Brownian motion will be given in §5.2.
The role of the infinitesimal generator is being played down here. For the one-dimensional case it is quite useful, see e.g., Ito [1] for some applications. In higher dimensions the domain of the operator is hard to describe and its full use is neither necessary nor sufficient for most purposes. It may be said that the substitution of integral operators (semigroup, resolvent, balayage) for differential ones constitutes an essential advance of the modern theory of Markov processes. Gauss and Koebe made the first fundamental step in identifying a harmonic function by its averaging property (Theorem 2 in §4.3). This is indeed a lucky event for probability theory.
This section is added as an afterthought to show that “there is still sap from the old tree”. For a more complete discussion see Chung and Rao [3] where D is not assumed to be bounded but m(D) ∞. The one-dimensional case is treated in Chung and Varadhan [1]. The functional e q (t) was introduced by Feynman with a purely imaginary q in his “path integrals”; by Kac [1] with a nonpositive q Its application to the Schrödinger equation is discussed in Dynkin [1] with q 0, Khas’minskii [1] with q > 0. The general case of a bounded q requires a new approach partly due to the lack of a maximum principle.
Let us alert the reader to the necessity of a meticulous verification of domination, such as given in (13), in the sort of calculations in Theorem 5. Serious mistakes have resulted from negligence on this score. For instance, it is not sufficient in this case to verify that u(x) ∞, as one might be misled to think after a preliminary (illicit) integration with respect t.
Comparison of the methods used here with the classical approach in elliptic partial differential equations should prove instructive. For instance, it can be shown that the finiteness of u(D,q, 1 ; •) in D is equivalent to the existence of a strictly positive solution belonging to C(2)(D) ∩ C(0)(D). This is also equivalent to the proposition that all eigenvalues λ of the Schrödinger operator, written in the form (Δ/2 + q)φ = λφ, are strictly negative; see a forthcoming paper by Chung and Li [1]. Further results are on the way.
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© 1982 Springer Science+Business Media New York
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Chung, K.L. (1982). Brownian Motion. 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_4
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DOI: https://doi.org/10.1007/978-1-4757-1776-1_4
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