Reaching the best possible rate of convergence to equilibrium for solutions of Kac’s equation via central limit theorem

Dolera, Emanuele; Gabetta, Ester; Regazzini, Eugenio

doi:10.1214/08-aap538

Cited by 28 publications

(40 citation statements)

References 25 publications

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“…In the case of the Kac equation, that has the Gaussian distribution as steady state, rates of convergence with respect to Kolmogorov's uniform metric, weighted χ-metrics of order p ≥ 2, Wasserstein metrics of order 1 and 2 and total variation distance have been proved. See [14,15,19]. As for the inelastic Kac equation, in [4] rates of convergence to equilibrium with respect to Kolmogorov's uniform metric and χ-weighted metrics have been derived.…”

Section: Introductionmentioning

confidence: 99%

Speed of convergence to equilibrium in Wasserstein metrics for Kac-like kinetic equations

Bassetti

Perversi

2013

Electron. J. Probab.

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This work deals with a class of one-dimensional measure-valued kinetic equations, which constitute extensions of the Kac caricature. It is known that if the initial datum belongs to the domain of normal attraction of an α-stable law, the solution of the equation converges weakly to a suitable scale mixture of centered α-stable laws. In this paper we present explicit exponential rates for the convergence to equilibrium in Kantorovich-Wasserstein distances of order p > α, under the natural assumption that the distance between the initial datum and the limit distribution is finite. For α = 2 this assumption reduces to the finiteness of the absolute moment of order p of the initial datum. On the contrary, when α < 2, the situation is more problematic due to the fact that both the limit distribution and the initial datum have infinite absolute moment of any order p > α. For this case, we provide sufficient conditions for the finiteness of the Kantorovich-Wasserstein distance.Date: November 12, 2018. Key words and phrases. Boltzmann-like equations and Kac caricature and stable laws and rate of convergence to equilibrium and Wasserstein distances .

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Section: Introductionmentioning

confidence: 99%

Speed of convergence to equilibrium in Wasserstein metrics for Kac-like kinetic equations

Bassetti

Perversi

2013

Electron. J. Probab.

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“…The general idea to represent solutions to Kac-like equations in a probabilistic way dates back at least to McKean [18]; this approach has been fully formalized and employed in the derivation of various analytical results in the last decade. For the original Kac equation, probabilistic methods have been used to estimate the approximation error of truncated Wild sums in [3], to study necessary and sufficient conditions for the convergence to a steady state in [12], to study the blow-up behavior of solutions of infinite energy in [4], to obtain rates of convergence to equilibrium of the solutions both in strong and weak metrics, [7,8,13]. Also the inelastic Kac model has been studied by probabilistic methods, see [2].…”

Section: Introductionmentioning

confidence: 99%

Central limit theorem for a class of one-dimensional kinetic equations

Bassetti

Ladelli

Matthes

2010

Probab. Theory Relat. Fields

135

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We introduce a class of kinetic-type equations on the real line, which constitute extensions of the classical Kac caricature. The collisional gain operators are defined by smoothing transformations with rather general properties. By establishing a connection to the central limit problem, we are able to prove long-time convergence of the equation's solutions toward a limit distribution. For example, we prove that if the initial condition belongs to the domain of normal attraction of a certain stable law ν α , then the limit is a scale mixture of ν α . Under some additional assumptions, explicit exponential rates for the convergence to equilibrium in Wasserstein metrics are calculated, and strong convergence of the probability densities is shown.

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“…The proof of the main theorem to be formulated in the next section rests on a probabilistic scheme originally given in [30,31] and exploited, for example, in [1,2,11,12,17,18,19,22,23,24]. Hence, we first touch on the basics of such a scheme.…”

Section: Preliminariesmentioning

confidence: 99%

Probabilistic View of Explosion in an Inelastic Kac Model

2014

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Let $\{\mu(\cdot,t):t\geq0\}$ be the family of probability measures corresponding to the solution of the inelastic Kac model introduced in Pulvirenti and Toscani [\textit{J. Stat. Phys.} \textbf{114} (2004) 1453-1480]. It has been proved by Gabetta and Regazzini [\textit{J. Statist. Phys.} \textbf{147} (2012) 1007-1019] that the solution converges weakly to equilibrium if and only if a suitable symmetrized form of the initial data belongs to the standard domain of attraction of a specific stable law. In the present paper it is shown that, for initial data which are heavier-tailed than the aforementioned ones, the limiting distribution is improper in the sense that it has probability 1/2 "adherent" to $-\infty$ and probability 1/2 "adherent" to $+\infty$. It is explained in which sense this phenomenon is amenable to a sort of explosion, and the main result consists in an explicit expression of the rate of such an explosion. The presentation of these statements is preceded by a discussion about the necessity of the assumption under which their validity is proved. This gives the chance to make an adjustment to a portion of a proof contained in the above-mentioned paper by Gabetta and Regazzini

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Reaching the best possible rate of convergence to equilibrium for solutions of Kac’s equation via central limit theorem

Cited by 28 publications

References 25 publications

Speed of convergence to equilibrium in Wasserstein metrics for Kac-like kinetic equations

Speed of convergence to equilibrium in Wasserstein metrics for Kac-like kinetic equations

Central limit theorem for a class of one-dimensional kinetic equations

Probabilistic View of Explosion in an Inelastic Kac Model

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