On a directionally reinforced random walk

Ghosh, Arka P.; Rastegar, Reza; Roitershtein, Alexander

doi:10.1090/s0002-9939-2014-12030-2

Cited by 14 publications

(13 citation statements)

References 36 publications

(63 reference statements)

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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%

“…Furthermore, it seems that some results in [25,27] are not entirely true when the persistence times are not integrable, and so it must have some misunderstanding in their proofs. We think more precisely to [ However, the latter convergence can not be true since Y α p1q is the overshoot limit of the underlying CTRW and thus it is not compactly supported -contrary to X α p1q.…”

Section: Around and Beyond Ctrwsmentioning

confidence: 99%

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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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“…For d = 1, we have two possible directions alternatively taken by the moving particle. The random flights have been studied, for instance, in [30,31,9,26,27,10,7,18,28]. Recently, in [14,15] the relationship between the isotropic transport processes and some fractional Klein-Gordon equations has been analyzed.…”

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confidence: 99%

Stochastic models associated to a Nonlocal Porous Medium Equation

Gregorio¹

2018

Modern Stochastics: Theory and Applications

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The nonlocal porous medium equation considered in this paper is a degenerate nonlinear evolution equation involving a space pseudo-differential operator of fractional order. This space-fractional equation admits an explicit, nonnegative, compactly supported weak solution representing a probability density function. In this paper we analyze the link between isotropic transport processes, or random flights, and the nonlocal porous medium equation. In particular, we focus our attention on the interpretation of the weak solution of the nonlinear diffusion equation by means of random flights. Applications is the fractional Laplace operator; i.e. a Fourier multiplier with the symbol ||ξ|| α . We observe that ∇ 1 = ∇ is the classical gradient and that div(Another equivalent definition of the fractional gradient ∇ α−1 involves the Riesz potential; that is ∇ α−1 = ∇I 2−α where I β = (−∆) − β 2 is a Fourier multiplier with symbol ||ξ|| −β , β ∈ (0, 2) (for more details on this point see [3,4]).In [3,4], explicit and compactly supported nonnegative self-similar solutions of (1) are constructed. These explicit solutions generalize the well-known Barenblatt-Kompanets-Zel'dovich-Pattle solutions of the porous medium equation (4) below. Furthermore, the authors proved the existence of sign-changing weak solution to the Cauchy problem (1)-(2) for u 0 (x) ∈ L 1 (R d ), and the hypercontractivity L 1 → L p estimates.By exploiting Darcy's law, it is possible to interpret the equation (1) as a transport equation ∂ t u = div(|u|v), where v := ∇p := ∇I 2−α (|u| m−2 u) is a vector velocity field with nonlocal and nonlinear pressure p in the case of nonnegative initial data. We observe that the fractional operator ∇I 2−α represents the long-range diffusion effects. The one-dimensional version of the pseudo-differential equation (1) describes the dynamics of dislocations in crystals (see [2]).For α = 2, (1) becomes the classical nonlinear porous medium equation

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On a directionally reinforced random walk

Cited by 14 publications

References 36 publications

Persistent Random Walks. I. Recurrence Versus Transience

Persistent Random Walks. I. Recurrence Versus Transience

Persistent Random Walks. II. Functional Scaling Limits

Stochastic models associated to a Nonlocal Porous Medium Equation

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