Convergence of moments for dispersing
                    billiards with cusps

Bálint, Péter; Chernov, N.; Dolgopyat, D.

doi:10.1090/conm/698/14033

Cited by 6 publications

(16 citation statements)

References 31 publications

(57 reference statements)

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“…Item (2) follows directly from Proposition 2 (2). This also implies that the total expansion factor during the interval [N 1 , N 3 ] is of order O(1).…”

Section: General Models With Cuspsmentioning

confidence: 66%

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Decay of correlations for billiards with flat points II: cusps effect

Zhang¹

2017

Contemporary Mathematics

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In this paper we continue to study billiards with flat points, by constructing a special family of dispersing billiards with cusps. All boundaries of the table have positive curvature except that the curvature vanishes at the vertex of cusps, i.e. the boundaries intersect at the flat point tangentially. We study the mixing rates of this one-parameter family of billiards parameterized by β ∈ (2, ∞), and show that the correlation functions of the collision map decay polynomially with order O(n

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“…Item (2) follows directly from Proposition 2 (2). This also implies that the total expansion factor during the interval [N 1 , N 3 ] is of order O(1).…”

Section: General Models With Cuspsmentioning

confidence: 66%

“…The rates were improved to O(n −1 ) in [14]. This model was further investigated in [1,2]. In [10], Chernov and Makarian also raised an open question: "It is interesting to let the curvature vanish at the vertex of the cusp, .... would this affect the rate of the decay of correlations?"…”

mentioning

confidence: 99%

Decay of correlations for billiards with flat points II: cusps effect

Zhang¹

2017

Contemporary Mathematics

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“…For a particle with phase point x = (r, v) ∈ M, let Φ t (x) give the phase point of the same particle after it moves for time t. For x = (r, v) ∈ M denote π Q x = r. {Φ t : M → M|t ∈ R} is called the billiard flow and this definition is unambiguous if we assume that, in addition, the trajectories are continuous from the right. 4 For x ∈ M, let τ (x) denote the time of free flight for x until the first collision:…”

Section: Billiard Tablementioning

confidence: 99%

Equidistribution for Standard Pairs in Planar Dispersing Billiard Flows

Bálint

Nándori

Szász

et al. 2018

Ann. Henri Poincaré

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We prove exponential correlation decay in dispersing billiard flows on the 2-torus assuming finite horizon and lack of corner points. With applications aimed at describing heat conduction, the highly singular initial measures are concentrated here on 1-dimensional submanifolds (given by standard pairs) and the observables are supposed to satisfy a generalized Hölder continuity property. The result is based on the exponential correlation decay bound of Baladi, Demers and Liverani [1] obtained for Hölder continuous observables in these billiards. The model dependence of the bounds is also discussed.describe the regularity of the billiard table Q for our purposes. In most of our calculations, constants that depend only on the billiard table Q, will actually depend on Q only through these regularity parameters. Note that the bounds τ min , τ max , κ min , κ max , κ ′ max , K max , d Q need not be sharp, so the estimates we give are uniform for the class of billiard tables satisfying the same bounds. Theorem 2.7 (Uniform hyperbolicity). There are constants λProof. In this whole section we assume that F : M → R is generalized α F -Hölder continuous (cf. Definition 2.12) and that M F dµ = 0. The analogous property for the billiard map is stated in Corollary 4.20 and Formula (4.19) in [12]. As formulated in Theorem 2.7, the statement follows from the definition of α min and the expansion properties of dispersing wave fronts, in particular Formulas (3.35) and (4.9) in [12]. See also Lemma 3.3 in [1].Theorem 2.8 (Transversality). There is a constant c tr = c tr (R Q , R u ) > 0 such that any u-curve and any central-stable manifold intersecting it have an angle at least c tr at their intersection point.

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“…Other examples of applications include certain kinds of transport in fluid flow [26,32], transport in biological cells [6,28], the migration of bacteria [1], predator search behavior [27], and traveling humans [7]. Also, Lorentz gases [16,33] and other billiards problems [2], in which there is no randomness in a strict sense but, rather, deterministic chaos, have been approximated by super-diffusive random walks.…”

mentioning

confidence: 99%

“…One can then use the Tauberian theorem to deduce the long-time asymptotics of the MSD from the behavior of derivatives with respect to the Fourier variable near 0 [19,21,29]. An alternative and more elementary approach is to express the MSD as an integral over the velocity auto-correlation, and evaluate or analyze the integral; see for instance [2,12]. We present a mathematical analysis of the basic properties of the MSD, using this approach, under the assumption that F has finite expectation.…”

mentioning

confidence: 99%

On the mean square displacement in Levy walks

Börgers¹,

Greengard²

2019

Preprint

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Many physical and biological processes are modeled by "particles" undergoing Lévy random walks. A feature of significant interest in these systems is the mean square displacement (MSD) of the particles. Long-time asymptotic approximations of the MSD have been established, via the Tauberian Theorem, for systems in which the distribution of the step durations is asymptotically a power law of infinite variance. We extend these results, using elementary analysis, and obtain closedform expressions as well as power law bounds for the MSD in equilibrium, and representations of the MSD as sums of super-linear, linear, and sub-linear terms. We show that the super-linear components are determined by the mean and asymptotics of the step durations, but that the linear and sub-linear components (whose size has implications for the accuracy of the asymptotic approximation) depend on the entire distribution function.

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Convergence of moments for dispersing billiards with cusps

Cited by 6 publications

References 31 publications

Decay of correlations for billiards with flat points II: cusps effect

Decay of correlations for billiards with flat points II: cusps effect

Equidistribution for Standard Pairs in Planar Dispersing Billiard Flows

On the mean square displacement in Levy walks

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