  An integral representation result for strictly positive subharmonic functions
of a one-dimensional regular diffusion is established. More precisely, any such
function can be written as a linear combination of an increasing and a
decreasing subharmonic function that solve an integral equation \[ g(x)=a +
\int v(x,y)\mu_A(dy) + \kappa s(x), \] where $a>0$, $\kappa \in \mathbb{R}$,
$s$ is a scale function of the diffusion, $\mu_A$ is a Radon measure, and $v$
is a kernel that is explicitly determined by the scale function. This integral
equation in turn allows one construct a pair $(g,A)$ such that $g$ is a
subharmonic function, $A$ is a continuous additive functional with Revuz
measure $\mu_A$ and $g(X)\exp(-A)$ is a local martingale. The changes of
measures associated with such pairs are studied and shown to modify the long
term behaviour of the original diffusion process to exhibit transience. Theory
is illustrated via examples that in particular contain a sequence of measure
transformations that render the diffusion irregular in the limit by breaking
the state space into distinct regions with soft and hard borders. Finally, the
theory is applied to find an "explicit" solution to an optimal stopping problem
with random discounting.
