Intrinsic small time estimates for distribution densities of L\'evy processes

Abstract

We construct intrinsic on-and off-diagonal upper and lower estimates for the transition probability density of a L\'evy process in small time. By intrinsic we mean that such estimates reflect the structure of the characteristic exponent of the process. The technique used in the paper relies on the asymptotic analysis of the inverse Fourier transform of the respective characteristic function. We provide several examples, in particular, with rather irregular L\'evy measure, to illustrate our results.