Quantitative propagation of chaos for generalized Kac particle systems

Cortez, Roberto; Fontbona, Joaquín

doi:10.1214/15-aap1107

Cited by 27 publications

(88 citation statements)

References 26 publications

(53 reference statements)

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“…samples, established in [11]). An interesting question, raised in [5], is to what extent this sub-optimality is intrinsic to the interaction type, or a consequence of the techniques employed.…”

Section: Comparison To Known Results and Approachesmentioning

confidence: 99%

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Quantitative Uniform Propagation of Chaos for Maxwell Molecules

Cortez

Fontbona

2018

Commun. Math. Phys.

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We prove propagation of chaos at explicit polynomial rates in Wasserstein distance W 2 for Kac's N -particle system associated with the spatially homogeneous Boltzmann equation for Maxwell molecules. Our approach is mainly based on novel probabilistic coupling techniques. Combining them with recent stabilization results for the particle system we obtain, under suitable moments assumptions on the initial distribution, a uniform-in-time estimate of order almost N −1/3 for W 2 2 .

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Section: Comparison To Known Results and Approachesmentioning

confidence: 99%

“…We are now ready to state and prove: Lemma 18. Assume (5) and that G N 0 ∈ P sym 2 ((R 3 ) N ) is concentrated on the Boltzmann sphere S N . Assume also that R p := sup N E|V 1 0 | p < ∞ for some p ≥ 4.…”

Section: Estimates and Technical Resultsmentioning

confidence: 99%

Quantitative Uniform Propagation of Chaos for Maxwell Molecules

Cortez

Fontbona

2018

Commun. Math. Phys.

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“…We now give a specific construction of the particle system and couple it with a suitable system of nonlinear processes, following [6]. Consider a Poisson point measure 2 : i(ξ) = i(ζ)} and i(ξ) := ⌊ξ⌋ + 1.…”

Section: Constructionmentioning

confidence: 99%

“…. , U N,t ) for (4) and (6) are straightforward: since the total rate of N is finite over finite time intervals, those equations are nothing but recursions for the values of the processes at the (timely ordered) jump times. Also, the collection of pairs (V 1 , U 1 ), .…”

Section: Constructionmentioning

confidence: 99%

Uniform Propagation of Chaos for Kac’s 1D Particle System

Cortez

2016

J Stat Phys

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In this paper we study Kac's 1D particle system, consisting of the velocities of N particles colliding at constant rate and randomly exchanging energies. We prove uniform (in time) propagation of chaos in Wasserstein distance with explicit polynomial rates in N , for both the squared (i.e., the energy) and non-squared particle system. These rates are of order N −1/3 (almost, in the non-squared case), assuming that the initial distribution of the limit nonlinear equation has finite moments of sufficiently high order (4 + ǫ is enough when using the 2-Wasserstein distance). The proof relies on a convenient parametrization of the collision recently introduced by Hauray, as well as on a coupling technique developed by Cortez and Fontbona.

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“…This latter bound is tight up to factors of log(n), and will be essential to our effort. Related Wasserstein bounds have also been found in [CF14] for several similar models. However, a mixing bound in the Wasserstein metric does not directly imply any mixing bound in the total variation metric.…”

Section: The Law Of {Vmentioning

confidence: 54%

On the mixing time of Kac’s walk and other high-dimensional Gibbs samplers with constraints

Pillai¹,

Smith²

2018

Ann. Probab.

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Determining the total variation mixing time of Kac's random walk on the special orthogonal group SO(n) has been a long-standing open problem. In this paper, we construct a novel non-Markovian coupling for bounding this mixing time. The analysis of our coupling entails controlling the smallest singular value of a certain random matrix with highly dependent entries. The dependence of the entries in our matrix makes it not-amenable to existing techniques in random matrix theory. To circumvent this difficulty, we extend some recent bounds on the smallest singular values of matrices with independent entries to our setting. These bounds imply that the mixing time of Kac's walk on the group SO(n) is between C 1 n 2 and C 2 n 4 log(n) for some explicit constants 0 < C 1 , C 2 < ∞, substantially improving on the bound of O(n 5 log(n) 2 ) in the preprint [Jia12b] of Jiang. Our methods may also be applied to other high dimensional Gibbs samplers with constraints and thus are of independent interest. In addition to giving analytical bounds on the mixing time, our approach allows us to compute rigorous estimates of the mixing time by simulating the eigenvalues of a random matrix. ‡

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Quantitative propagation of chaos for generalized Kac particle systems

Cited by 27 publications

References 26 publications

Quantitative Uniform Propagation of Chaos for Maxwell Molecules

Quantitative Uniform Propagation of Chaos for Maxwell Molecules

Uniform Propagation of Chaos for Kac’s 1D Particle System

On the mixing time of Kac’s walk and other high-dimensional Gibbs samplers with constraints

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