We study the small deviation problem logℙ(sup _{t∈[0, 1]}|X_t|≤ɛ), as ɛ→0, for general Lévy processes X. The techniques enable us to determine the asymptotic rate for general real-valued Lévy processes, which we demonstrate with many examples.

As a particular consequence, we show that a Lévy process with nonvanishing Gaussian component has the same (strong) asymptotic small deviation rate as the corresponding Brownian motion.