On the local rate of growth of Lévy processes with no positive jumps (Q1346146)

Let \(\Phi\) be the right inverse of the Lévy exponent \(\Psi\) of a Lévy process \(X_ t\) starting at the origin: \(E\exp(aX_ t)= \exp(t\Psi(a))\). The main results extend what hold for stable processes with suitable index. They are: \(X_ t\) without positive jumps, then \[ \limsup_{t\to 0} \frac{(X_{\rho+ t}- X_ t)\Phi (t^{-1} \log|\log t|)}{\log|\log t|}= c> 0\quad\text{a.s.}, \] where \(\rho\) stands for the a.s. unique instant of the minimum of \(X\) on \(t\in [0,1]\), and \[ \limsup_{t\to 0} \frac{(X_{\tau+ t}- X_ t) \Phi(t^{-1}|\log t|)} {|\log t|}\geq {1\over 2}\quad\text{for some }\tau\quad\text{a.s.}, \] while \(\leq 6\) for all \(\tau\) a.s., which shows the existence of fast points of \(X\).