  For a L\'evy process $\xi=(\xi_t)_{t\geq0}$ drifting to $-\infty$, we define
the so-called exponential functional as follows
\[{\rm{I}}_{\xi}=\int_0^{\infty}e^{\xi_t} dt.\] Under mild conditions on $\xi$,
we show that the following factorization of exponential functionals
\[{\rm{I}}_{\xi}\stackrel{d}={\rm{I}}_{H^-} \times {\rm{I}}_{Y}\] holds, where,
$\times $ stands for the product of independent random variables, $H^-$ is the
descending ladder height process of $\xi$ and $Y$ is a spectrally positive
L\'evy process with a negative mean constructed from its ascending ladder
height process. As a by-product, we generate an integral or power series
representation for the law of ${\rm{I}}_{\xi}$ for a large class of L\'evy
processes with two-sided jumps and also derive some new distributional
properties. The proof of our main result relies on a fine Markovian study of a
class of generalized Ornstein-Uhlenbeck processes which is of independent
interest on its own. We use and refine an alternative approach of studying the
stationary measure of a Markov process which avoids some technicalities and
difficulties that appear in the classical method of employing the generator of
the dual Markov process.
