Convergence of random walks with Markovian cookie stacks to Brownian motion perturbed at extrema

Kosygina, Elena; Mountford, Thomas; Peterson, Jonathon

doi:10.1007/s00440-021-01055-3

Cited by 6 publications

(9 citation statements)

References 25 publications

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“…A few of the above models are of the type where the (recurrent 4 ) random walk after visiting a site "in the bulk" (i.e., away from the boundary of its current range) a certain fixed number of times makes only unbiased steps from that site 5 , and for these models the convergence to a BMPE seems very intuitive. On the other hand, for excited random walks with Markovian cookie stacks the work [KMP22]) established the convergence to a multiple of BMPE, even though the Brownian behavior "in the bulk" is not seen at the random walk level. Establishing this behavior is non-trivial and requires intermediate "mesoscopic" coarse graining of space and time.…”

Section: Introduction and Main Resultsmentioning

confidence: 98%

“…This raised the possibility of finding a general result whereby a functional limit theorem for the rescaled walk could simply be proven via establishing the Ray-Knight theorems and minimal technical conditions. A number of results for excited random walks, [DK12, KP16,KMP22], weighed positively on this possibility except that the "technical conditions" remained elusive and the treatment varied significantly from model to model.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…The restriction to p > 1 2 guarantees that the series representing the "total drift" at a single site (see (11) and Lemma 2.3) converges absolutely. We think that the case p ∈ (0, 1/2] can be considered as well but will certainly require a different and technically more involved treatment such as coarse graining in the spirit of [KMP22].…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

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Convergence and non-convergence of scaled self-interacting random walks to Brownian motion perturbed at extrema

Kosygina¹,

Mountford²,

Peterson³

2022

Preprint

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We use generalized Ray-Knight theorems introduced by B. Tóth in [Tót96] together with techniques developed for excited random walks as main tools for establishing positive and negative results concerning convergence of some classes of diffusively scaled self-interacting random walks (SIRWs) to Brownian motions perturbed at extrema (BMPE). The work [Tót96] studied two classes of SIRWs: asymptotically free and polynomially self-repelling walks. For both classes B. Tóth has shown, in particular, that the distribution function of a scaled SIRW observed at independent geometric times converges to that of a BMPE indicated by the generalized Ray-Knight theorem for this SIRW. The question of weak convergence of one-dimensional distributions of scaled SIRW remained open. In this paper, on the one hand, we prove a full functional limit theorem for a large class of asymptotically free SIRWs which includes asymptotically free walks considered in [Tót96]. On the other hand, we show that rescaled polynomially self-repelling SIRWs do not converge to the BMPE predicted by the corresponding generalized Ray-Knight theorems and, hence, do not converge to any BMPE.

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Section: Introduction and Main Resultsmentioning

confidence: 98%

Section: Introduction and Main Resultsmentioning

confidence: 99%

Section: Introduction and Main Resultsmentioning

confidence: 99%

See 1 more Smart Citation

Convergence and non-convergence of scaled self-interacting random walks to Brownian motion perturbed at extrema

Kosygina¹,

Mountford²,

Peterson³

2022

Preprint

Self Cite

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“…where this last inequality follows from (7). Using the fact that Φ is 1 √ 2π −Lipschitz, we can bound this expression by…”

Section: Substituting This Expression For E[t N ] Back Into the First...mentioning

confidence: 99%

“…These models have the additional complication that δ, as expressed above in (1), may not even be well-defined. Recurrence/transience criteria, law of large numbers, and functional limit theorems have been proven for both models [7][8][9], but the results are given in terms of parameters that are associated to ERW through branching-like processes, a generalization of the branching processes used for environments with finitely many cookies which we will discuss in a later section. These parameters can be computed explicitly, but their formulas are somewhat complicated.…”

Section: Introductionmentioning

confidence: 99%

Recurrence/Transience criteria for excited random walks with finite-drift cookie stacks

Letterhos¹

2021

Preprint

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We consider excited random walk (ERW) on Z in environments with identical stacks of infinitely many cookies at each site, subject to the constraint that the total drift per site δ = (2p j − 1) is finite. Building on the methods of Kozma, Orenshtein, and Shinkar [11], we show that ERW in finite-drift environments is recurrent when |δ| < 1 and transient when |δ| > 1. In the case |δ| = 1 we prove that ERW is recurrent under mild assumptions on the environment. In addition, we show that ERW may be transient when |δ| = 1, an interesting new behavior that was not present in previously studied models.

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Speed of Excited Random Walks with Long Backward Steps

Nguyen

2022

J Stat Phys

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We study a model of multi-excited random walk with non-nearest neighbour steps on $$\mathbb {Z}$$ Z , in which the walk can jump from a vertex x to either $$x+1$$ x + 1 or $$x-i$$ x - i with $$i\in \{1,2,\dots ,L\}$$ i ∈ { 1 , 2 , ⋯ , L } , $$L\ge 1$$ L ≥ 1 . We first point out the multi-type branching structure of this random walk and then prove a limit theorem for a related multi-type Galton–Watson process with emigration, which is of independent interest. Combining this result and the method introduced by Basdevant and Singh (Probab Theory Relat Fields 141:3–4, 2008), we extend their result (w.r.t. the case $$L=1$$ L = 1 ) to our model. More specifically, we show that in the regime of transience to the right, the walk has positive speed if and only if the expected total drift $$\delta >2$$ δ > 2 . This confirms a special case of a conjecture proposed by Davis and Peterson.

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Convergence of random walks with Markovian cookie stacks to Brownian motion perturbed at extrema

Cited by 6 publications

References 25 publications

Convergence and non-convergence of scaled self-interacting random walks to Brownian motion perturbed at extrema

Convergence and non-convergence of scaled self-interacting random walks to Brownian motion perturbed at extrema

Recurrence/Transience criteria for excited random walks with finite-drift cookie stacks

Speed of Excited Random Walks with Long Backward Steps

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