On Dean–Kawasaki Dynamics with Smooth Drift Potential

Konarovskyi, Vitalii; Lehmann, Tobias; Renesse, Max von

doi:10.1007/s10955-019-02449-3

Cited by 20 publications

(8 citation statements)

References 33 publications

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“…Formally, the latter is a stochastic version of a standard Fokker-Planck equation (of order 1 or 2 depending on the cases) that includes an additional noisy term whose local quadratic variation derives exactly from the Riemannian metric on P(R d ) (with d = 1 in [41,42,66]). However, it has been proved in [44,45] that the Dean-Kawasaki equation, in its strict version, cannot be solvable except in trivial cases where it reduces to a finite dimensional particle system (which requires in particular the initial distribution to be finitely supported). This negative result has an interesting consequence: to make the Dean-Kawasaki equation well-posed, some extra correction is needed in the dynamics, which is exactly what is done in [41,42,66].…”

Section: Diffusions With Values In the Space Of Probability Measuresmentioning

confidence: 99%

Rearranged Stochastic Heat Equation

Delarue¹,

Hammersley²

2022

Preprint

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The purpose of this work is to provide an explicit construction of a strong Feller semigroup on the space of probability measures over the real line that additionally maps bounded measurable functions into Lipschitz continuous functions, with a Lipschitz constant that blows up in an integrable manner in small time. Our construction relies on a rearranged version of the stochastic heat equation on the circle driven by a coloured noise. Formally, this stochastic equation writes as a reflected equation in infinite dimension, a topic that is known to be challenging. Under the action of the rearrangement, the solution is forced to live in a space of quantile functions that is isometric to the space of probability measures on the real line. We prove the equation to be solvable by means of an Euler scheme in which we alternate flat dynamics in the space of random variables on the circle with a rearrangement operation that projects back the random variables onto the subset of quantile functions. A first challenge is to prove that this scheme is tight. A second one is to provide a consistent theory for the limiting reflected equation and in particular to interpret in a relevant manner the reflection term. The last step in our work is to establish the aforementioned Lipschitz property of the semigroup by adapting earlier ideas from the Bismut-Elworthy-Li formula in stochastic analysis.

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Section: Diffusions With Values In the Space Of Probability Measuresmentioning

confidence: 99%

Rearranged Stochastic Heat Equation

Delarue¹,

Hammersley²

2022

Preprint

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“…The basis of this dichotomy is the interplay of the particular geometry of diffusion and noise in the context of a stochastic Wasserstein gradient flow. We also mention that a similar setting later led the authors of [22] to obtain an analogous dichotomy in the case of more general smooth drift potentials F [23]. The central differences to the approach presented below are that in [22], the underlying particle dynamics is first order (overdamped Langevin); the noise is derived from deep probabilistic arguments (describing Brownian motion in the space of probability measures with finite second moment, i.e., relying on the Wasserstein geometry); and the noise is not regularised.…”

Section: Introductionmentioning

confidence: 71%

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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“…Subsequently, alternative constructions have been given in [1] and [52]. Negative results on the existence to the unmodified Dean-Kawasaki equation with space-time white Itô noise have been recently obtained in [50,51]. A regularized model replacing the Dean-Kawasaki equation was derived and analyzed by Cornalba, Shardlow and Zimmer in [14,15], by means of smoothed particles with second order (underdamped) Langevin dynamics.…”

Section: Comments On the Results And Assumptionsmentioning

confidence: 99%

Well-posedness of the Dean--Kawasaki and the nonlinear Dawson--Watanabe equation with correlated noise

Fehrman¹,

Gess²

2021

Preprint

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In this paper we prove the well-posedness of the generalized Dean-Kawasaki equation driven by noise that is white in time and colored in space. The results treat diffusion coefficients that are only locally 1 /2-Hölder continuous, including the square root. This solves several open problems, including the well-posedness of the Dean-Kawasaki equation and the nonlinear Dawson-Watanabe equation with correlated noise.

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On Dean–Kawasaki Dynamics with Smooth Drift Potential

Abstract: We consider the Dean-Kawasaki equation with smooth drift interaction potential and show that measure valued solutions exist only in certain parameter regimes in which case they are given by finite Langevin particle systems with mean field interaction.

Cited by 20 publications

References 33 publications

Rearranged Stochastic Heat Equation

Rearranged Stochastic Heat Equation

A Regularized Dean--Kawasaki Model: Derivation and Analysis

Well-posedness of the Dean--Kawasaki and the nonlinear Dawson--Watanabe equation with correlated noise

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