Modified Massive Arratia Flow and Wasserstein Diffusion

Konarovskyi, Vitalii; Renesse, Max von

doi:10.1002/cpa.21758

Cited by 39 publications

(44 citation statements)

References 41 publications

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“…It finishes the proof of the theorem. [21] give that Y (·, t), t ∈ [0, T ], is an L ↑ 2 [0, 1]-valued continuous square integrable martingale with the quadratic variation (M ′ ). Thus, the conditions (C1) − (C4) are equivalent (M) or (M ′ ).…”

Section: 2mentioning

confidence: 99%

“…The modified Arratia flow has a connection with the Wasserstein diffusion, constructed by M.-K. von Renessse and T. Sturm in [31] (see also [2,29,30]). In fact, in [21] V. Konarovskyi and M.-K. von Renesse proved that the process describing the evolution of particle mass in the modified Arratia flow solves a SPDE that is similar to the SPDE for the Wasserstein diffusion and also showed via a large deviation analysis that the flow satisfies the Varadhan formula with the square of the Wasserstein distance as the rate function. 1], then the process…”

Section: Introductionmentioning

confidence: 99%

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On asymptotic behavior of the modified Arratia flow

Konarovskyi¹

2017

Electron. J. Probab.

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We study asymptotic properties of the system of interacting diffusion particles on the real line which transfer a mass [arXiv:1408.0628]. The system is a natural generalization of the coalescing Brownian motions. The main difference is that diffusion particles coalesce summing their mass and changing their diffusion rate inversely proportional to the mass. First we construct the system in the case where the initial mass distribution has the moment of the order greater then two as an $L_2$-valued martingale with a suitable quadratic variation. Then we find the relationship between the asymptotic behavior of the particles and local properties of the mass distribution at the initial time

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Section: 2mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

On asymptotic behavior of the modified Arratia flow

Konarovskyi¹

2017

Electron. J. Probab.

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“…In order to estimate ct, we need to compute E[p 2 (t)w (x − q(t))]. We again rely on the conditional law (21) and the law of total expectation to write…”

Section: Proof Of Proposition 11 Under Assumption (G)mentioning

confidence: 99%

“…Thirdly, the lack of Lipschitz continuity associated with the square root poses further difficulties. Von Renesse and collaborators have studied regularised versions of (1) in the foundational works [33,3,20,21]. They obtain existence results for measure-valued martingale solutions for modifications of (1) (in [3,33] for the Gibbs-Boltzmann entropy functional F scaled by µ > 0, and in [20] for the case F ≡ 0).…”

Section: Introductionmentioning

confidence: 99%

A Regularized Dean--Kawasaki Model: Derivation and Analysis

Cornalba¹,

Shardlow²,

Zimmer³

2019

SIAM J. Math. Anal.

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The Dean-Kawasaki model consists of a nonlinear stochastic partial differential equation featuring a conservative, multiplicative, stochastic term with non-Lipschitz coefficient, and driven by space-time white noise; this equation describes the evolution of the density function for a system of finitely many particles governed by Langevin dynamics. Well-posedness for the Dean-Kawasaki model is open except for specific diffusive cases, corresponding to overdamped Langevin dynamics. There, it was recently shown by Lehmann, Konarovskyi, and von Renesse that no regular (non-atomic) solutions exist.We derive and analyse a suitably regularised Dean-Kawasaki model of wave equation type driven by coloured noise, corresponding to second order Langevin dynamics, in one space dimension. The regularisation can be interpreted as considering particles of finite size rather than describing them by atomic measures. We establish existence and uniqueness of a solution. Specifically, we prove a highprobability result for the existence and uniqueness of mild solutions to this regularised Dean-Kawasaki model. Proposition 1.1 (Tightness of {ρ } , {j } , {j 2, } ). Let T > 0, and let D ⊂ R be a bounded domain. Assume the validity of either Assumption (G) or Assumption (NG), given below in Subsection 2.2. Then the families of processes of {ρ } , {j } are tight in C(0, T ; L 2 (D)) and C(0, T ; L 4 (D)), respectively, for N θ ≥ 1, with θ ≥ 3. In addition, the family {j 2, } is tight in C(0, T ; L 4 (D)) for N θ ≥ 1, with θ ≥ 5. Proposition 1.1 yields relative compactness in law for the families of processes {ρ } , {j } , {j 2, } as → 0. We show convergence for the family {ρ } as → 0 in the following result. Proposition 1.2. Let T > 0, and let D ⊂ R be a bounded domain. Assume the validity of either Assumption (G) or Assumption (NG), as well as the scaling N θ ≥ 1, for some θ ≥ 3. For each > 0, let η be the law of the process ρ on X := C(0, T ; L 2 (D)). There exists a probability measure η on X such that η w → η in X as → 0. Here w → denotes weak convergence of measures.The proofs of Proposition 1.1 and 1.2 under Assumption (G) are the content of Subsection 3.1.

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“…The Arratia flow introduced in [1] provides an example of this kind. In recent years, it has been being extensively studied, see [4,7,8,11] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

Random jumps and coalescence in the continuum: Evolution of states of an infinite particle system

Kozitsky¹,

Pilorz²

2020

Discrete &Amp; Continuous Dynamical Systems - A

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The dynamics of an infinite continuum system of randomly jumping and coalescing point particles is studied. The states of the system are probability measures on the corresponding configuration space Γ the evolution of which is constructed in the following way. The evolution of observables F0 → Ft is obtained from a Kolmogorovtype evolution equation. Then the evolution of states µ0 → µt is defined by the relation µ0(Ft) = µt(F0) for F0 belonging to a measure-defining class of functions. The main result of the paper is the proof of the existence of the evolution of this type for a bounded time horizon.2010 Mathematics Subject Classification. 60K35; 82C22.

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Modified Massive Arratia Flow and Wasserstein Diffusion

Cited by 39 publications

References 41 publications

On asymptotic behavior of the modified Arratia flow

On asymptotic behavior of the modified Arratia flow

A Regularized Dean--Kawasaki Model: Derivation and Analysis

Random jumps and coalescence in the continuum: Evolution of states of an infinite particle system

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