Small deviations of general L\'{e}vy processes

We study the small deviation problem $\log \mathbb{P}({\sup }_{t\in [0,1]}\mathrm{\mid }{X}_t\mathrm{\mid }\le \epsilon )$, as $\epsilon \to 0$, for general L'{e}vy processes $X$. The techniques enable us to determine the asymptotic rate for general real-valued L'{e}vy processes, which we demonstrate with many examples. As a particular consequence, we show that a L'{e}vy process with nonvanishing Gaussian component has the same (strong) asymptotic small deviation rate as the corresponding Brownian motion.