Persistent random walks. II. Functional Scaling Limits
Abstract
We give a complete and unified description  under some stability assumptions  of the functional scaling limits associated with some persistent random walks for which the recurrent or transient type is studied in [1]. As a result, we highlight a phase transition phenomenon with respect to the memory. It turns out that the limit process is either Markovian or not according to  to put it in a nutshell  the rate of decrease of the distribution tails corresponding to the persistent times. In the memoryless situation, the limits are classical strictly stable L{é}vy processes of infinite variations. However, we point out that the description of the critical Cauchy case fills some lacuna even in the closely related context of Directionally Reinforced Random Walks (DRRWs) for which it has not been considered yet. Besides, we need to introduced some relevant generalized drift  extended the classical one  in order to study the critical case but also the situation when the limit is no longer Markovian. It appears to be in full generality a drift in mean for the Persistent Random Walk (PRW). The limit processes keeping some memory  given by some variable length Markov chain  of the underlying PRW are called arcsine Lamperti anomalous diffusions due to their marginal distribution which are computed explicitly here. To this end, we make the connection with the governing equations for L{é}vy walks, the occupation times of skew Bessel processes and a more general class modelled on Lamperti processes. We also stress that we clarify some misunderstanding regarding this marginal distribution in the framework of DRRWs. Finally, we stress that the latter situation is more flexible  as in the first paper  in the sense that the results can be easily generalized to a wider class of PRWs without renewal pattern.
 Publication:

arXiv eprints
 Pub Date:
 December 2016
 arXiv:
 arXiv:1612.00238
 Bibcode:
 2016arXiv161200238C
 Keywords:

 Mathematics  Probability