The entrance laws of self-similar Markov processes and exponential functionals of Lévy processes
A Markov process \(X_t\) on \((0,\infty)\) with law \(\mathbb P^{x}\), \(x>0\), is called semi-stable (or self-similar) with index \(1/\alpha\), if \(\text{law}(kX_{k^{-\alpha}t}, t\geq 0) = \mathbb P^{kx}\), \(k,x>0\). Lamperti established a one-to-one correspondence between semi-stable Markov processes \(X_t\) and Lévy processes \(\xi_t\) which involves time-changes using an exponential functional of the Lévy process. The present paper investigates the limiting behaviour of \(\mathbb P^x\) as \(x\downarrow 0\). This question comes up quite naturally when one wants to characterize the class of all Markov processes which can arise in limit theorems [cf. \textit{J. Lamperti}, Z. Wahrscheinlichkeitstheorie Verw. Geb. 22, 205-225 (1972; Zbl 0274.60052)], but it is also linked, via the scaling property, to the long-time asymptotics of \(X_t\): \(\lim_{x\to 0} \mathbb P^x \circ X_1^{-1} = \mathbb P^0\circ X^{-1}_1\) weakly if and only if weak-\(\lim_{t\to\infty} \mathbb P^y\circ(t^{-1/\alpha}X_t)^{-1! } = \mathbb P^0\circ X_1^{-1}\). The main result of the paper shows that \(\mathbb P^x \to \mathbb P^0\) as \(x\downarrow 0\) in the sense of finite-dimensional distributions whenever the Lévy process \(\xi_t\) is not of lattice type and has strictly positive means. In this case, the semigroup associated with \(X_t\) is Feller, i.e., acts on the space of continuous functions on \((0,\infty)\) which vanish at infinity. This result extends a recent paper by \textit{J. Bertoin} and \textit{M.-E. Caballero} [Bernoulli 8, 195-205 (2002; Zbl 1002.60032)] where the Lévy process was assumed to be a subordinator. The proof of the theorem is based on the renewal theorem and duality arguments which are well-known for the Lévy process \(\xi_t\) and which are extended to the semi-stable Markov process \(X_t\).
scientific article; zbMATH DE number 1790689