A Wiener-Hopf Type Factorization for the Exponential Functional of Levy Processes

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.