×

Criteria of weak and strong transience for Lévy processes. (English) Zbl 0963.60043

Watanabe, S. (ed.) et al., Probability theory and mathematical statistics. Proceedings of the seventh Japan-Russia symposium, Tokyo, Japan, July 26-30, 1995. Singapore: World Scientific. 438-449 (1996).
A stochastic process on \(\mathbb{R}^n\) with semigroup \((P_s)_{s\geq 0}\) is called weakly (strongly) transient if it is transient and if \[ \int^\infty_0 \int^\infty_t \int h\cdot P_s hdx ds dt<\infty \quad(= \infty) \] for some \(h\in C^+_c (\mathbb{R}^n)\). This notion is due to S. C. Port [J. Combinat. Theory 2, 107-128 (1967; Zbl 0162.49101)]. Criteria for transience and recurrence of Lévy processes are well-known (Chung-Fuchs-type criteria, Spitzer-type criteria). The author shows that a transient Lévy process with characteristic exponent \(\psi\) which is strongly non-lattice, i.e. \(\lim\sup_{|z|\to \infty}|e^{\psi(z)} |<1\), is weakly (strongly) transient according to \[ \limsup_{\lambda\to 0}\int_{|z|< \varepsilon} \text{Re} {1\over(\lambda -\psi(z))^2}dz= \infty\text{ or }< \infty \] for some \(\varepsilon>0\). This is – up to the square in the numerator of the integrand – the Chung-Fuchs-type criterion for transience. Using the example of a real-valued, asymmetric stable process of order 1 and a Brownian motion with drift, one can see that a Spitzer-type criterion for weak (strong) transience is not available. The proofs rely heavily on the interplay between Lévy processes and Fourier transforms (Plancherel’s theorem).
For the entire collection see [Zbl 0939.00029].

MSC:

60G51 Processes with independent increments; Lévy processes
60G17 Sample path properties
60J35 Transition functions, generators and resolvents

Citations:

Zbl 0162.49101
PDFBibTeX XMLCite