Recurrent extensions of self-similar Markov processes and Cramér's condition II

Abstract

We prove that a positive self-similar Markov process $(X,\mathbb{P})$ that hits 0 in a finite time admits a self-similar recurrent extension that leaves 0 continuously if and only if the underlying Lévy process satisfies Cramér's condition.

arXiv commentPublished in at http://dx.doi.org/10.3150/07-BEJ6082 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm)

References in corpus (3)

• Exponential functionals of Levy processes
• Recurrent extensions of self-similar Markov processes and Cramér's condition II
• Weak convergence of positive self-similar Markov processes and overshoots of Lévy processes

Cited by in corpus (33)

• Recurrent extensions of self-similar Markov processes and Cramér's condition II
• Weak convergence of positive self-similar Markov processes and overshoots of Lévy processes
• The hitting time of zero for a stable process
• Hitting distributions of alpha-stable processes via path censoring and self-similarity
• On the distribution of exponential functionals for Levy processes with jumps of rational transform
• A Wiener-Hopf Type Factorization for the Exponential Functional of Levy Processes
• Probabilistic aspects of critical growth-fragmentation equations
• Fluctuation theory and exit systems for positive self-similar Markov processes
• Distributional properties of exponential functionals of Levy processes
• On the Lamperti stable processes
• A Jump Type SDE Approach to Positive Self-Similar Markov Processes
• A transformation for Lévy processes with one-sided jumps and applications
• Law of the exponential functional of a new family of one-sided Levy processes via self-similar continuous state branching processes with immigration and the Wright hypergeometric functions
• Fluctuations of stable processes and exponential functionals of hypergeometric Levy processes
• Infinite divisibility of solutions to some self-similar integro-differential equations and exponential functionals of Lévy processes
• On the density of exponential functionals of Lévy processes
• The extended hypergeometric class of Lévy processes
• Intertwining, Excursion Theory and Krein Theory of Strings for Non-self-adjoint Markov Semigroups
• Precise asymptotics for the density and the upper tail of exponential functionals of subordinators
• Hitting properties and non-uniqueness for SDE driven by stable processes
• Asymptotic behaviour of first passage time distributions for Lévy processes
• Entrance laws at the origin of self-similar Markov processes in high dimensions
• On continuous state branching processes: conditioning and self-similarity
• A temporal factorization at the maximum for spectrally negative positive self-similar Markov processes
• Functional limit theorems for processes pieced together from excursions
• Path decomposition of a spectrally negative Lévy process, and local time of a diffusion in this environment
• On entire moments of self-similar Markov processes
• Spectral decomposition of fractional operators and a reflected stable semigroup
• Recurrent extensions of real-valued self-similar Markov processes
• On the exact asymptotics of exit time from a cone of an isotropic $\alpha$-self-similar Markov process with a skew-product structure
• Intertwining certain fractional derivatives
• A Ciesielski-Taylor type identity for positive self-similar Markov processes
• Extended Factorizations of Exponential Functionals of Lévy Processes