  We develop a new class of path transformations for one-dimensional diffusions
that are tailored to alter their long-run behaviour from transient to recurrent
or vice versa. This immediately leads to a formula for the distribution of the
first exit times of diffusions, which is recently characterised by Karatzas and
Ruf \cite{KR} as the minimal solution of an appropriate Cauchy problem under
more stringent conditions. A particular limit of these transformations also
turn out to be instrumental in characterising the stochastic solutions of
Cauchy problems defined by the generators of strict local martingales, which
are well-known for not having unique solutions even when one restricts
solutions to have linear growth. Using an appropriate diffusion transformation
we show that the aforementioned stochastic solution can be written in terms of
the unique classical solution of an {\em alternative} Cauchy problem with
suitable boundary conditions. This in particular resolves the long-standing
issue of non-uniqueness with the Black-Scholes equations in derivative pricing
in the presence of {\em bubbles}. Finally, we use these path transformations to
propose a unified framework for solving explicitly the optimal stopping problem
for one-dimensional diffusions with discounting, which in particular is
relevant for the pricing and the computation of optimal exercise boundaries of
perpetual American options.
