Potential theory on infinite products and locally compact groups

It is well-known that postulating a few properties of a sheaf of continuous 'harmonic' functions is enough to obtain most of the results of classical potential theory. Two main sets of axioms emerged from this development: Brelot axiomatic theory, which covers the classical elliptic theory, and Bauer axiomatic theory which also applies to parabolic equations. In the elliptic case, Bauer theory is weaker than Brelot theory: Any Brelot harmonic sheaf is a Bauer elliptic harmonic sheaf. In view of the translation invariance of the classical theory, it is natural to study invariant sheaves of harmonic functions on groups. Let E be a connected and locally connected locally compact group having a countable basis for its topology. Does E admit a translation invariant Brelot harmonic sheaf? For which E does the elliptic Bauer theory coincide with Brelot theory for all invariant harmonic sheaves? This paper settles these questions and shows that: (a) any E carries invariant Brelot harmonic sheaves; (b) any invariant elliptic Bauer harmonic sheaf is a Brelot sheaf if and only if E is a finite dimensional Lie group. To obtain these results we construct certain convolution semigroups of Gaussian measures having prescribed analytic properties such as: (a) a continuous Green kernel off the diagonal; or (b) a Green function which is absolutely continuous with respect to the Haar measure but whose excessive (lower semi-continuous) density has a dense set of poles off the diagonal. The proofs depend on structure theorems for connected, locally connected, locally compact groups and semisimple Lie groups. Because of the structure of these groups, the crucial case is that of an infinite product of compact Lie groups. The key ingredient in the proofs is a set of analytical results for product diffusions on infinite products of manifolds obtained in a companion paper. Our results give a satisfactory description of the analytic and potential theoretic behavior of the natural diffusions on any infinite product of semisimple compact Lie groups. They generalize Berg's and Bendikov's results concerning T-infinity. Other cases such as products of spheres are also considered.