Approximation rate in Wasserstein distance of probability measures on the real line by deterministic empirical measures

Bencheikh, Oumaima; Jourdain, Benjamin

doi:10.1016/j.jat.2021.105684

Cited by 5 publications

(6 citation statements)

References 9 publications

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“…To obtain existence as well as approximation of solutions to the generalized Vlasov equation, let us first investigate properties of F β . To construct approximations of the digraph measures as well as the initial distribution ν 0 , we rely on a result for approximation of probability measures by finitely supported probability measures [40,12] (see also [5,26]) under the bounded Lipschitz metric. Furthermore, we will use the parametrized metrics d I,α BL induced by the bounded Lipschitz metric.…”

Section: Generalized Vementioning

confidence: 99%

“…x m i ,s m,n,0 of η x m i ,s 0 by finitely supported measures follow from [40,12,5] (see also [26,Lemma 5.6]): There exists a sequence {y m,n (i−1)n+j : j = 1, . .…”

Section: Appendix a Gronwall Inequalitiesmentioning

confidence: 99%

“…To do this, we rely on the continuous dependence of the semiflow (Propositions 4.1 and 4.6). Such an approximation result is also based on the discretization of a given initial digraph measure as well as the initial distribution (i.e., the initial condition of the generalized VE), which further is a consequence of the recent results of probability measures by finitely supported discrete measures with equal mass on each point of the support (so-called uniform approximation [40], or deterministic empirical approximation [12,5]) [40,12,5]. We point out that except for the existence of the approximations of the initial digraph measure, we also need to ensure the sequence of graphs of the approximations are nonnegative over the prescribed finite time interval of interest.…”

mentioning

confidence: 99%

“…Here "digraph measure" refers a measure-valued function, regarded as a generalization of graphon as a graph limit[26] 5. η is understood as η(A × B) = A η x (B)dµ X (x) for A, B ∈ B(X)[26].…”

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confidence: 99%

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Mean field limits of co-evolutionary heterogeneous networks

Gkogkas¹,

Kuehn²,

Xu³

2022

Preprint

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Many science phenomena are modelled as interacting particle systems (IPS) coupled on static networks. In reality, network connections are far more dynamic. Connections among individuals receive feedback from nearby individuals and make changes to better adapt to the world. Hence, it is reasonable to model myriad real-world phenomena as co-evolutionary (or adaptive) networks. These networks are used in different areas including telecommunication, neuroscience, computer science, biochemistry, social science, as well as physics, where Kuramoto-type networks have been widely used to model interaction among a set of oscillators. In this paper, we propose a rigorous formulation for limits of a sequence of co-evolutionary Kuramoto oscillators coupled on heterogeneous co-evolutionary networks, which receive feedback from the dynamics of the oscillators on the networks. We show under mild conditions, the mean field limit (MFL) of the co-evolutionary network exists and the sequence of co-evolutionary Kuramoto networks converges to this MFL. Such MFL is described by solutions of a generalized Vlasov type equation. We treat the graph limits as graph measures, motivated by the recent work in [Kuehn, Xu. Vlasov equations on digraph measures, arXiv:2107.08419, 2021]. Under a mild condition on the initial graph measure, we show that the graph measures are positive over a finite time interval. In comparison to the recently emerging works on MFLs of IPS coupled on non-co-evolutionary networks (i.e., static networks or time-dependent networks independent of the dynamics of the IPS), our work is the first to rigorously address the MFL of a co-evolutionary network model. The approach is based on our formulation of a generalization of the co-evolutionary network as a hybrid system of ODEs and measure differential equations parametrized by a vertex variable, together with an analogue of the variation of parameters formula, as well as the generalized Neunzert's in-cell-particle method developed in [Kuehn, Xu. Vlasov equations on digraph measures, arXiv:2107.08419, 2021].

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Section: Generalized Vementioning

confidence: 99%

“…x m i ,s m,n,0 of η x m i ,s 0 by finitely supported measures follow from [40,12,5] (see also [26,Lemma 5.6]): There exists a sequence {y m,n (i−1)n+j : j = 1, . .…”

Section: Appendix a Gronwall Inequalitiesmentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

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Mean field limits of co-evolutionary heterogeneous networks

Gkogkas¹,

Kuehn²,

Xu³

2022

Preprint

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“…Then, under rather mild conditions for the DHGM (basically, uniform boundedness of the DHGM), we obtain the well-posedness of the VE (a transport type PDE whose uniform weak solution is the density of the MFL). To obtain the approximation of the MFL by empirical measures, we rely on the recently developed result of uniform approximation of probability measures on Euclidean spaces [44,12,3] (see also [27]), which is different from the Martingale Convergence Theorem for the approximation of integrable functions exmployed in [24]. For the approximation results to hold, we need some continuity of the DHGM.…”

Section: Introductionmentioning

confidence: 99%

Vlasov Equations on Directed Hypergraph Measures

Kuehn¹,

Xu²

2022

Preprint

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In this paper we propose a framework to investigate the mean field limit (MFL) of interacting particle systems on directed hypergraphs. We provide a non-trivial measuretheoretic viewpoint and make extensions of directed hypergraphs as directed hypergraph measures (DHGMs), which are measure-valued functions on a compact metric space. These DHGMs can be regarded as hypergraph limits which include limits of a sequence of hypergraphs that are sparse, dense, or of intermediate densities. Our main results show that the Vlasov equation on DHGMs are well-posed and its solution can be approximated by empirical distributions of large networks of higher-order interactions. The results are applied to a Kuramoto network in physics, an epidemic network, and an ecological network, all of which include higher-order interactions. To prove the main results on the approximation and well-posedness of the Vlasov equation on DHGMs, we robustly generalize the method of [Kuehn, Xu. Vlasov equations on digraph measures, arXiv:2107.08419, 2021] to higherdimensions. In particular, we successfully extend the arguments for the measure-valued functions f : X → M + (X) to those for f : X → M + (X k−1 ), where X is the vertex space of DHGMs and k ∈ N \ {1} is the cardinality of the DHGM.

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Vlasov equations on digraph measures

Kuehn

2022

Journal of Differential Equations

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Approximation rate in Wasserstein distance of probability measures on the real line by deterministic empirical measures

Cited by 5 publications

References 9 publications

Mean field limits of co-evolutionary heterogeneous networks

Mean field limits of co-evolutionary heterogeneous networks

Vlasov Equations on Directed Hypergraph Measures

Vlasov equations on digraph measures

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