We prove several necessary and sufficient conditions for the existence of (smooth) transition probability densities for Lévy processes and isotropic Lévy processes. Under some mild conditions on the characteristic exponent we calculate the asymptotic behaviour of the transition density as t → 0 and t → ∞ and show a ratio-limit theorem.MSC 2010: Primary: 60G51. Secondary: 60E10, 60F99, 60J35.Abstract. We prove several necessary and sufficient conditions for the existence of (smooth) transition probability densities for Lévy processes and isotropic Lévy processes. Under some mild conditions on the characteristic exponent we calculate the asymptotic behaviour of the transition density as t → 0 and t → ∞ and show a ratio-limit theorem.MSC 2010: Primary: 60G51. Secondary: 60E10, 60F99, 60J35.

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We construct intrinsic on-and off-diagonal upper and lower estimates for the transition probability density of a Lévy 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 Lévy measure, to illustrate our results.

Abstract. We study for a class of symmetric Lévy processes with state space R n the transition density p t (x) in terms of two one-parameter families of metrics, (d t ) t>0 and (δ t ) t>0 . The first family of metrics describes the diagonal term p t (0); it is induced by the characteristic exponent ψ of the Lévy process by d t (x, y) = tψ(x − y). The second and new family of metrics δ t relates to √ tψ through the formulawhere F denotes the Fourier transform. Thus we obtain the following "Gaussian" representation of the transition density:t (x,0) where p t (0) corresponds to a volume term related to √ tψ and where an "exponential" decay is governed by δ 2 t . This gives a complete and new geometric, intrinsic interpretation of p t (x). MSC 2010: Primary: 60J35. Secondary: 60E07; 60E10; 60G51; 60J45; 47D07; 31E05.

We show on-and off-diagonal upper estimates for the transition densities of symmetric Lévy and Lévy-type processes. To get the on-diagonal estimates we prove a Nash type inequality for the related Dirichlet form. For the off-diagonal estimates we assume that the characteristic function of a Lévy (type) process is analytic, which allows to apply the complex analysis technique.