Density behaviour related to Lévy processes

Abstract: Let $p_t(x)$, $f_t(x)$ and $q_t^*(x)$ be the densities at time $t$ of a real L\'evy process, its running supremum and the entrance law of the reflected excursions at the infimum. We provide relationships between the asymptotic behaviour of $p_t(x)$, $f_t(x)$ and $q_t^*(x)$, when $t$ is small and $x$ is large. Then for large $x$, these asymptotic behaviours are compared to this of the density of the L\'evy measure. We show in particular that, under mild conditions, if $p_t(x)$ is comparable to $t\nu(x)$, as $t\rightarrow0$ and $x\rightarrow\infty$, then so is $f_t(x)$.