Separability and completeness for the Wasserstein distance

Bolley, François

doi:10.1007/978-3-540-77913-1_17

Cited by 79 publications

(69 citation statements)

References 4 publications

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“…The velocity u(t, t (x)) is the velocity at time t of a material point that started in x at time 0, and -in words - (4) means that this velocity is equal to the time derivative at time t of the position t (x) of that particular material point. By connecting the Eulerian velocity u to the Lagrangian velocity˙ t , we obtain the Lagrangian functional…”

Section: Derivation Of the Action In A Continuous Settingmentioning

confidence: 99%

“…Assumption (37) on convergence of initial data: Given the initial probability measure μ 0 supported in the ball B(r 0 ), we demonstrate here two ways of constructing an approximating sequence of measures (μ n 0 ) n∈N + . The first way of constructing μ n 0 is deterministic and has been used in [4]. For simplicity of presentation, we assume…”

Section: Proof Of Part 1 Of Theorem 311mentioning

confidence: 99%

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From continuum mechanics to SPH particle systems and back: Systematic derivation and convergence

et al. 2017

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In this paper, we derive from the principle of least action the equation of motion for a continuous medium with regularized density field in the context of measures. The eventual equation of motion depends on the order in which regularization and the principle of least action are applied. We obtain two different equations, whose discrete counterparts coincide with the scheme used traditionally in the Smoothed Particle Hydrodynamics (SPH) numerical method [27], and with the equation treated by Di Lisio et al. in [9], respectively. Additionally, we prove the convergence in the Wasserstein distance of the corresponding measure-valued evolutions, moreover providing the order of convergence of the SPH method. The convergence holds for a general class of force fields, including external and internal conservative forces, friction and non-local interactions. The proof of convergence is illustrated numerically by means of one and two-dimensional examples.

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Section: Derivation Of the Action In A Continuous Settingmentioning

confidence: 99%

Section: Proof Of Part 1 Of Theorem 311mentioning

confidence: 99%

From continuum mechanics to SPH particle systems and back: Systematic derivation and convergence

et al. 2017

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“…We start by summarizing the main properties of the Euclidean Wasserstein distance W 2 that we shall make use of in the rest, refering to [11,27] for the proofs. Proposition 1.…”

Section: Contraction In W 2 Of the Gain Operatormentioning

confidence: 99%

Tanaka Theorem for Inelastic Maxwell Models

Bolley

Carrillo

2007

Commun. Math. Phys.

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Abstract. We show that the Euclidean Wasserstein distance is contractive for inelastic homogeneous Boltzmann kinetic equations in the Maxwellian approximation and its associated Kac-like caricature. This property is as a generalization of the Tanaka theorem to inelastic interactions. Even in the elastic classical Boltzmann equation, we give a simpler proof of the Tanaka theorem than the ones in [25,27]. Consequences are drawn on the asymptotic behavior of solutions in terms only of the Euclidean Wasserstein distance.

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“…Our next result shows that such a construction is indeed possible. Its proof is based on the proof made in [5] to establish the separability of (P 1 (Y), W 1 ).…”

Section: Notation and Preliminary Resultsmentioning

confidence: 99%

Computable approximations for continuous-time Markov decision processes on Borel spaces based on empirical measures

Anselmi

Dufour²,

Prieto-Rumeau³

2016

Journal of Mathematical Analysis and Applications

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In this paper, we propose an approach for approximating the value function and an -optimal policy of continuous-time Markov decision processes with Borel state and action spaces, with possibly unbounded cost and transition rates, under the total expected discounted cost optimality criterion. Under adequate assumptions, which in particular include that the transition rate has a density function with respect to a reference measure, together with piecewise Lipschitz continuity of the elements of the control model, we approximate the original controlled process by a model with finite state and action spaces. The approximation error is related to the 1-Wasserstein distance between suitably defined probability measures and approximating measures with finite support. We also study the case when the reference measure is approximated with empirical distributions and we show that convergence of the approximations takes place at an exponential rate in probability.

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Separability and completeness for the Wasserstein distance

Cited by 79 publications

References 4 publications

From continuum mechanics to SPH particle systems and back: Systematic derivation and convergence

From continuum mechanics to SPH particle systems and back: Systematic derivation and convergence

Tanaka Theorem for Inelastic Maxwell Models

Computable approximations for continuous-time Markov decision processes on Borel spaces based on empirical measures

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