Potential kernel, hitting probabilities and distributional asymptotics

Pène, Françoise; Thomine, Damien

doi:10.1017/etds.2018.136

Cited by 8 publications

(23 citation statements)

References 57 publications

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“…for some finite set I and some injective function σ : I → R d . This kind of limit theorems is in line with a few former studies [32,40]. In this setting, hitting probabilities could be obtained in [40] for families of subsets of size 2.…”

Section: Introductionsupporting

confidence: 83%

“…This kind of limit theorems is in line with a few former studies [32,40]. In this setting, hitting probabilities could be obtained in [40] for families of subsets of size 2. However, these results were byproducts of distributional limit theorems for additive functionals of the process ( T n ) n≥0 .…”

Section: Introductionsupporting

confidence: 83%

“…By [40,Lemma 3.17], if the minimal distance between distinct sites in Σ ε increases to +∞ when ε vanishes, then the transitions from one site to another becomes increasingly unlikely, so that lim ε→0 P ε = Id. In this article, we are interested in the next order asymptotics of the family (P ε ) ε>0 when ε vanishes.…”

Section: Asymptotics Of the Transition Matrices And Irreductibilitymentioning

confidence: 99%

“…Since the probability of hitting [0] before going back to [p] converges to 0 as p goes to infinity by [40,Lemma 4.17], the random variable ϕ [p]→[0] converges in distribution to +∞ when p goes to infinity. In particular, there exists r 0 > 0 such that µ(ϕ [p]→[0] < n 0 ) < δ 0 /n 0 whenever p > r 0 .…”

Section: Definition 31 (Color Of a Site)mentioning

confidence: 99%

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Potential theory and $\mathbb{Z}^d$-extensions

Thomine¹

2021

Preprint

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We study hitting probabilities for Z d -extensions of Gibbs-Markov maps. The goal is to estimate, given a finite Σ ⊂ Z d and p, q ∈ Σ, the probability P pq that the process starting from p returns to Σ at site q.Our study generalizes the methods available for random walks. We are able to give in many settings (square integrable jumps, jumps in the basin of a Lévy or Cauchy random variable) asymptotics for the transition matrix (P pq ) p,q∈Σ when the elements of Σ are far apart.We use three main tools: a variant of the balayage identity using a transfer operator as a Markov transition kernel, a study inspired from fast-slow systems and the hitting time of small sets in hyperbolic systems to relate transfer operators and the transition matrices we seek to compute, and finally Fourier transform and perturbations of transfer operators à la Nagaev-Guivarc'h to effectively compute these transition matrices in an asymptotic regime.

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Section: Introductionsupporting

confidence: 83%

Section: Introductionsupporting

confidence: 83%

Section: Asymptotics Of the Transition Matrices And Irreductibilitymentioning

confidence: 99%

Section: Definition 31 (Color Of a Site)mentioning

confidence: 99%

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Potential theory and $\mathbb{Z}^d$-extensions

Thomine¹

2021

Preprint

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“…Still in dynamical contexts, another approach based on moments has been developed in [41,42] in parallel to the coupling method. This second method based on local limit theorem is well tailored to treat non-markovian situations, such as RWRS.…”

Section: Resultsmentioning

confidence: 99%

Limit theorems for additive functionals of random walks in random scenery

Pène¹

2021

Electron. J. Probab.

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We study the asymptotic behaviour of additive functionals of random walks in random scenery. We establish bounds for the moments of the local time of the Kesten and Spitzer process. These bounds combined with a previous moment convergence result (and an ergodicity result) imply the convergence in distribution of additive observables (with a normalization in n 1 4 ). When the sum of the observable is null, the previous limit vanishes and we prove the convergence in the sense of moments (with a normalization in n 1 8 ).

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Local Limit Theorem for Randomly Deforming Billiards

Demers

Pène

Zhang

2020

Commun. Math. Phys.

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We study limit theorems in the context of random perturbations of dispersing billiards in finite and infinite measure. In the context of a planar periodic Lorentz gas with finite horizon, we consider random perturbations in the form of movements and deformations of scatterers. We prove a Central Limit Theorem for the cell index of planar motion, as well as a mixing Local Limit Theorem for the cell index with piecewise Hölder continuous observables. In the context of the infinite measure random system, we prove limit theorems regarding visits to new obstacles and self-intersections, as well as decorrelation estimates. The main tool we use is the adaptation of anisotropic Banach spaces to the random setting.

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Potential kernel, hitting probabilities and distributional asymptotics

Cited by 8 publications

References 57 publications

Potential theory and $\mathbb{Z}^d$-extensions

Potential theory and $\mathbb{Z}^d$-extensions

Limit theorems for additive functionals of random walks in random scenery

Local Limit Theorem for Randomly Deforming Billiards

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