Topics in self-interacting random walks

Rastegar, Reza

doi:10.31274/etd-180810-1196

Cited by 3 publications

(3 citation statements)

References 43 publications

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“…Besides, as regards the scaling limits of DRRW, we refer to [20][21][22] where are revealed diffusive and super-diffusive behaviours. We expect in a forthcoming paper to extend these results to the asymmetric situation and fill some gaps left open.…”

Section: Related Resultsmentioning

confidence: 99%

Persistent Random Walks. I. Recurrence Versus Transience

Cénac

Loynes

et al. 2016

J Theor Probab

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We consider a walker on the line that at each step keeps the same direction with a probability which depends on the discrete time already spent in the direction the walker is currently moving. More precisely, the associated left-infinite sequence of jumps is supposed to be a Variable Length Markov Chain (VLMC) built from a probabilized context tree given by a double-infinite comb. These walks with memories of variable length can be seen as generalizations of Directionally Reinforced Random Walks (DRRW) introduced in [1, Mauldin & al., Adv. Math., 1996] in the sense that the persistence times are anisotropic. We give a complete characterization of the recurrence and the transience in terms of the probabilities to persist in the same direction or to switch. We point out that the underlying VLMC is not supposed to admit any stationary probability. Actually, the most fruitful situations emerge precisely when there is no such invariant distribution. In that case, the recurrent and the transient property are related to the behaviour of some embedded random walk with an undefined drift so that the asymptotic behaviour depends merely on the asymptotics of the probabilities of change of directions unlike the other case in which the criterion reduces to a drift condition. Finally, taking advantage of this flexibility, we give some (possibly random and lacunar) perturbations results and treat the case of more general probabilized context trees built by grafting subtrees onto the double-infinite comb.

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Section: Related Resultsmentioning

confidence: 99%

Persistent Random Walks. I. Recurrence Versus Transience

Cénac

Loynes

et al. 2016

J Theor Probab

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“…Those are nearest neighbourhood random walks on Z d keeping their directions during random times τ, independently and identically drawn after every change of directions. We refer to [25][26][27] where are revealed diffusive and super-diffusive behaviours. These walks are intrinsically continuous and can be seen as a linear interpolation of a CTRW -as the "true" Lévy walks studied in [15].…”

Section: Around and Beyond Ctrwsmentioning

confidence: 99%

Persistent Random Walks. II. Functional Scaling Limits

Cénac

Loynes

et al. 2018

J Theor Probab

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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 Lévy 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 Lévy 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.

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“…It would be interested to extend the results of Rastegar (2012) for the maximum local time to the PRWCRE. It seems plausible that the proof methods of Rastegar (2012) which rely on the branching-random walk duality can be carried over to the model considered in this thesis. This would require having in hand delicate estimates for exit times and exit probabilities, similar to those obtained for the classical ERW in Kosygina and…”

Section: Chapter 4 Conclusion and Future Directionsmentioning

confidence: 99%

Limit theorems for persistent random walks in cookie environments

Voller

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Topics in self-interacting random walks

Cited by 3 publications

References 43 publications

Persistent Random Walks. I. Recurrence Versus Transience

Persistent Random Walks. I. Recurrence Versus Transience

Persistent Random Walks. II. Functional Scaling Limits

Limit theorems for persistent random walks in cookie environments

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