Suppose that $E$ is an arbitrary domain in $\mathbb{R}^d$, $L$ is a
second order elliptic differential operator in $S = \mathbb{R}_+ \times E$ and
$S^e$ is the extremal part of the Martin boundary for the corresponding
diffusion $\xi$. Let $1 < \alpha \leq 2$. We investigate a boundary value
problem
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involving two measures $\eta$ and $\nu$. For the
existence of a solution, we give sufficient conditions in terms of a Martin
capacity and necessary conditions in terms of hitting probabilities for an $(L,
\alpha)$-superdiffusion $X$. If a solution exists, then it can be expressed by
an explicit formula through an additive functional $A$ of $X$.
¶ An $(L, \alpha)$-superdiffusion is a branching measure-valued
process. A natural linear additive (NLA) functional $A$ of $X$ is determined
uniquely by its potential $h$ defined by the formula $P_{\mu} A(0, \infty) =
\int h(r, x) \mu (dr, dx)$ for all $\mu \in \mathscr{M}^*$ (the determining set
of $A$). Every potential $h$ is an exit rule for $\xi$ and it has a unique
decomposition into extremal exit rules. If $\eta$ and $\nu$ are measures which
appear in this decomposition, then (*)can be replaced by an integral equation
$$\tag{**} u(r, x) + \int p(r, x; t, dy)u(t,
y)^{\alpha} ds = h(r, x),$$
where $p(r, x; t, dy)$ is the transition function of
$\xi$. We prove that h is the potential of a NLA functional if and only if (**)
has a solution $u$. Moreover,
$$u(r, x) = -\log P_{r, x} e^{-A(0, \infty)}.$$
¶ By applying these results to homogeneous functionals of
time-homogeneous superdiffusions, we get a stronger version of theorems proved
in an earlier publication. The foundation for our present investigation is laid
by a general theory developed in the accompanying paper.