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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 $$f,g\in L^1$$) $$T_tf(x)/T_{t+s}g(x) \overset{t\to\infty}{\longrightarrow}\int f d z / \int g d z$$.

##### MSC:
 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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