We investigate densities of vaguely continuous convolution semigroups of probability measures on the Euclidean space. We expose that many typical conditions on the characteristic exponent repeatedly used in the literature of the subject are equivalent to the behaviour of the maximum of the density as a function of time variable. We also prove qualitative lower estimates under mild assumptions on the corresponding jump measure and the characteristic exponent.
"Lévy processes: Concentration function and heat kernel bounds." Bernoulli 26 (4) 3191 - 3223, November 2020. https://doi.org/10.3150/20-BEJ1220