Hitting times of points for symmetric Lévy processes with completely monotone jumps

Abstract

Small-space and large-time estimates and asymptotic expansion of the distribution function and (the derivatives of) the density function of hitting times of points for symmetric Lévy processes are studied. The Lévy measure is assumed to have completely monotone density function, and a scaling-type condition $\mathrm{inf} \xi \Psi"(\xi) / \Psi'(\xi) > 0$ is imposed on the Lévy-Khintchine exponent $\Psi$. Proofs are based on generalised eigenfunction expansion for processes killed upon hitting the origin.