Classical potential theory can be roughly characterized as the
study of Newtonian potentials and the Laplace operator on the
Euclidean space JR3. It was discovered around 1930 that there is a
profound connection between classical potential 3 theory and the
theory of Brownian motion in JR . The Brownian motion is determined
by its semigroup of transition probabilities, the Brownian
semigroup, and the connection between classical potential theory
and the theory of Brownian motion can be described analytically in
the following way: The Laplace operator is the infinitesimal
generator for the Brownian semigroup and the Newtonian potential
kernel is the" integral" of the Brownian semigroup with respect to
time. This connection between classical potential theory and the
theory of Brownian motion led Hunt (cf. Hunt 2]) to consider
general "potential theories" defined in terms of certain stochastic
processes or equivalently in terms of certain semi groups of
operators on spaces of functions. The purpose of the present
exposition is to study such general potential theories where the
following aspects of classical potential theory are preserved: (i)
The theory is defined on a locally compact abelian group. (ii) The
theory is translation invariant in the sense that any translate of
a potential or a harmonic function is again a potential,
respectively a harmonic function; this property of classical
potential theory can also be expressed by saying that the Laplace
operator is a differential operator with constant co efficients."
General
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