×

zbMATH — the first resource for mathematics

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
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Blumenthal R. M., J. Math. Mech. 10 pp 493– (1961)
[2] Bodnarchuk S., Theory Probab. Math. Statist. 79 pp 20– (2008)
[3] DOI: 10.1515/FORM.2001.51 · Zbl 0978.47031 · doi:10.1515/FORM.2001.51
[4] DOI: 10.4064/dm393-0-1 · Zbl 0990.46018 · doi:10.4064/dm393-0-1
[5] DOI: 10.2307/2371683 · Zbl 0063.01951 · doi:10.2307/2371683
[6] DOI: 10.1112/plms/s3-38.2.335 · Zbl 0401.60069 · doi:10.1112/plms/s3-38.2.335
[7] Hoh W., Osaka J. Math. 35 pp 798– (1998)
[8] DOI: 10.1214/aop/1176994308 · Zbl 0526.60061 · doi:10.1214/aop/1176994308
[9] DOI: 10.1214/aoms/1177698325 · Zbl 0172.22101 · doi:10.1214/aoms/1177698325
[10] DOI: 10.1016/j.spa.2006.10.003 · Zbl 1118.60037 · doi:10.1016/j.spa.2006.10.003
[11] Schenk W., Wiss. Z. TU Dresden 24 pp 945– (1975)
[12] DOI: 10.1007/s10959-009-0208-8 · Zbl 1393.60050 · doi:10.1007/s10959-009-0208-8
[13] DOI: 10.1002/mana.200711116 · Zbl 1194.47044 · doi:10.1002/mana.200711116
[14] DOI: 10.1090/S0002-9947-1965-0182061-4 · doi:10.1090/S0002-9947-1965-0182061-4
[15] Volevich L. R., Uspechi Mat. Nauk 121 pp 3– (1965)
[16] DOI: 10.1007/s004400050011 · Zbl 04560628 · doi:10.1007/s004400050011
[17] Zabczyk J., Studia Math. 35 pp 227– (1970)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.