We study the small deviation problem logℙ(sup t∈[0, 1]|Xt|≤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.
"Small deviations of general Lévy processes." Ann. Probab. 37 (5) 2066 - 2092, September 2009. https://doi.org/10.1214/09-AOP457