Brelot harmonic space

Roughly speaking, a Brelot harmonic space is a locally compact space endowed with an additional structure making it possible to study notions known from classical potential theory, such as harmonic and hyperharmonic functions, potentials, minimum principle, the Dirichlet problem, harmonic measure, balayage, fine topology, Martin compactification, etc. Standard examples are provided by elliptic partial differential equations in Euclidean spaces or on manifolds or by harmonic functions on a Riemann surface. (Cf. also Harmonic space; Potential theory, abstract.)

Let $ X $ be a locally compact, locally connected topological space and let $ {\mathcal H} $ be a sheaf of vector spaces of real-valued continuous functions. This means that to every non-empty open set $ U \subset X $, a vector space $ {\mathcal H} _ {U} $ consisting of continuous functions on $ U $ is associated in such a way that: i) if $ f \in {\mathcal H} _ {U} $, $ V \subset U $ is a non-empty open set, then the restriction $ f \mid _ {V} \in {\mathcal H} _ {V} $; and ii) if $ {\mathcal U} \neq \emptyset $ is a family of non-empty open sets with union $ V $ and $ f $ is a function on $ V $ such that $ f \mid _ {U} \in {\mathcal H} _ {U} $ for every $ U \in {\mathcal U} $, then $ f \in {\mathcal H} _ {V} $. (The elements of $ {\mathcal H} _ {U} $ are called harmonic functions on $ U $ with respect to $ {\mathcal H} $; cf. also Harmonic function.)

The sheaf $ {\mathcal H} $ is called a Brelot harmonic structure (the terminology from [a1]) if the following three axioms hold:

I) $ {\mathcal H} $ is not degenerate, i.e., for every $ x \in X $ there exists an open neighbourhood $ U $ of $ x $ and a strictly positive function $ h \in {\mathcal H} _ {U} $;

II) (the base axiom) the topology of $ X $ has a basis consisting of regular sets. (Here, a set $ V \subset X $ is said to be regular (with respect to $ {\mathcal H} $) if the Dirichlet problem on $ V $ is solvable in the following sense: For every real-valued continuous function $ f $ on the boundary of $ V $, there exists a uniquely determined harmonic function $ H _ {f} \in {\mathcal H} _ {V} $ which extends $ f $ continuously. Furthermore, $ f \geq 0 $ implies $ H _ {f} \geq 0 $.)

III) the Brelot convergence axiom: for every increasing sequence $ ( h _ {n} ) $ of harmonic functions on a domain $ U \subset X $ one has $ \sup \{ {h _ {n} } : {n \in \mathbf N } \} \in {\mathcal H} _ {U} $, provided that $ \sup \{ {h _ {n} ( x ) } : {n \in \mathbf N } \} < \infty $ for some $ x \in U $.

The pair $ ( X, {\mathcal H} ) $ is then called a Brelot harmonic space. (These spaces were introduced by M. Brelot in 1957; see [a3]. Later on, more general axiomatic settings for potential theory were developed, mainly by H. Bauer, C. Constantinescu and A. Cornea, J. Bliedtner and W. Hansen, N. Boboc, Gh. Bucur and Cornea; see [a1] and [a2].)

Results of J.M. Bony (see e.g. [a1]) show that, in a sense, the theory of Brelot harmonic spaces is close to potential theory for second-order partial differential equations of elliptic type (cf. also Potential theory, abstract). A great deal of results known from classical potential theory (which corresponds to the Laplace equation) can be obtained in the framework of Brelot harmonic spaces; sometimes additional hypotheses are imposed. Moreover, there is an important and deep connection between a class of Markov processes and Brelot harmonic spaces (see [a1] and [a2] and Markov process).

References

[a1]	H. Bauer, "Harmonic spaces; a survey" Conf. Sem. Mat. Univ. Bari , 197 (1984)
[a2]	J. Bliedtner, W. Hansen, "Potential theory; an analytic and probabilistic approach to balayage" , Springer (1986)
[a3]	M. Brelot, "Axiomatique des fonctions harmoniques" , Presses Univ. Montréal (1966)