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A note on the existence of transition probability densities of Lévy processes. (English) Zbl 1269.60050
Let \(\left(p_t\right)\) denote the transition probabilities of a Lévy process – i.e., a continuous convolution semigroup – on \(\mathbb{R}^n\) with Lévy Khinchin representation \(\widehat{p}_t = \mathrm{e}^{-t\psi}\), \(\psi\) determined by the triple \((\ell, Q, \nu)\), denoting the drift term, Gaussian term and Lévy measure, respectively. The classical criteria of P. Hartmann and A. Wintner [Am. J. Math. 64, 273–298 (1942; Zbl 0063.01951)] give necessary and sufficient conditions for the existence of \(C^\infty\)-densities (denoted by \(p_t(\cdot)\)). Precisely, if \(\lim_{|\xi|\to\infty} \operatorname{Re} \psi(\xi)/\ln(1+|\xi|) =\infty\) (\(HW_\infty\)), then there exist densities \(p_t\in L^1\cap C^\infty\), and, if \(\mathrm{liminf}_{|\xi|\to\infty}\operatorname{Re} \psi(\xi)/\ln(1+|\xi|)>n/t\) (\(HW_{1/t}\)), then there exist (smooth) densities \(p_s\in L^1\cap C^\infty\) for \(s\geq t\).
The aim of the investigations is to characterize processes for which (\(HW_\infty\)) is necessary and sufficient for the existence of smooth densities. The main results (Theorems 2.1 and 2.2) present conditions for Lévy processes without Gaussian component which are equivalent to (\(HW_\infty\)). In particular, for isotropic processes, (\(HW_\infty\)) is equivalent to \(\mathrm{e}^{-t \psi}\in L^1\) (for all \(t\)). Furthermore, then (\(HW_\infty\)) is equivalent to \(\lim_{\epsilon\to 0} \nu( B(0,\epsilon)^C)/|\ln(\epsilon)| =\infty\).
The results are illustrated by a series of examples. In Section 4, the results are extended, considering \(HW\)-conditions for the decreasing re-arrangement of \(\operatorname{Re}\psi\) and comparison with with the characteristic exponent \(\phi\) of a further Lévy process. The corresponding \(HW\)-condition is obtained replacing \(\ln(1+|\xi|)\) (in (\(HW_\infty\)) ) by \(\ln(1+\phi(\xi))\). This yields (Theorem 4.2) – under the assumption \((1+\phi)^{-\kappa/2}\in L^2\) (for some \(\kappa>0\)) – further equivalent conditions guaranteeing the existence of smooth densities.
Section 5 is concerned with some applications, e.g., in Theorem 5.7, quotient ergodic theorems are obtained for the transition \(C_0\)-transition semigroup \((T_t)\), as \( T_tf(x)/||\mathrm{e}^{-t\psi}||_{L^1}\overset{t\to\infty}{\longrightarrow} (1/2\pi)^n \int f d z\) (\(f\in L^1\)), or, (for \(f,g\in L^1\)) \(T_tf(x)/T_{t+s}g(x) \overset{t\to\infty}{\longrightarrow}\int f d z / \int g d z\).

60G51 Processes with independent increments; Lévy processes
60E07 Infinitely divisible distributions; stable distributions
60E10 Characteristic functions; other transforms
60F99 Limit theorems in probability theory
60J35 Transition functions, generators and resolvents
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