Density estimate in small time for jump processes with singular Lévy measures

Abstract

We consider the asymptotic behaviour of the transition density for processes of jump type as the time parameter $t$ tends to 0. We use Picard's duality method, which allows us to obtain the lower and upper bounds of the density even for the case where the support of Lévy measure is singular. The main result is that, under certain restrictions, the density behaves in polynomial order or may decrease in exponential order as $t\to0$ according to geometrical conditions of the objective points.