Deterministic and stochastic hydrodynamic equations arising from simple microscopic model systems

Giacomin, Giambattista; Lebowitz, Joel; Presutti, Errico

doi:10.1090/surv/064/03

Cited by 78 publications

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“…Another example is the dynamical Φ 4 3 model arising for example in the stochastic quantisation of Euclidean quantum field theory [PW81, JLM85, AR91, DPD03, Hai14], as well as a universal model for phase coexistence near the critical point [GLP99]:…”

Section: Introductionmentioning

confidence: 99%

Introduction to Regularity Structures

Friz

Hairer²

2014

Universitext

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These are short notes from a series of lectures given at the University of Rennes in June 2013, at the University of Bonn in July 2013, at the XVII th Brazilian School of Probability in Mambucaba in August 2013, and at ETH Zurich in September 2013. They give a concise overview of the theory of regularity structures as exposed in the article [Hai14]. In order to allow to focus on the conceptual aspects of the theory, many proofs are omitted and statements are simplified. We focus on applying the theory to the problem of giving a solution theory to the stochastic quantisation equations for the Euclidean Φ 4 3 quantum field theory.

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Section: Introductionmentioning

confidence: 99%

Introduction to Regularity Structures

Friz

Hairer²

2014

Universitext

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“…The proof follows the case-by-case reasoning in Section 6 for (15). The error terms R n,j , j = 1, 2, 3, are the same as there.…”

Section: Proof Of Theorem 24mentioning

confidence: 91%

“…Let us emphasize that the limits in (15) and (19) are valid for any finite collection of points, including shock locations. For the global results, (16) and part (ii) of Theorem 2.2, we integrate over space so that the values of the processes at shocks become immaterial because the shocks are a Lebesgue null set.…”

Section: Remarkmentioning

confidence: 99%

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Diffusive Fluctuations for One-Dimensional Totally Asymmetric Interacting Random Dynamics

Seppäläinen

2002

Commun. Math. Phys.

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We study central limit theorems for a totally asymmetric, one-dimensional interacting random system. The models we work with are the Aldous-Diaconis-Hammersley process and the related stick model. The A-D-H process represents a particle configuration on the line, or a 1-dimensional interface on the plane which moves in one fixed direction through random local jumps. The stick model is the process of local slopes of the A-D-H process, and has a conserved quantity. The results describe the fluctuations of these systems around the deterministic evolution to which the random system converges under hydrodynamic scaling. We look at diffusive fluctuations, by which we mean fluctuations on the scale of the classical central limit theorem. In the scaling limit these fluctuations obey deterministic equations with random initial conditions given by the initial fluctuations. Of particular interest is the effect of macroscopic shocks, which play a dominant role because dynamical noise is suppressed on the scale we are working.

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“…A characteristic property of Appell sequences is the following identity (see for instance [2], the equation between (14) and Remark 3 therein with X distributed according to the measure μ here and Y = 0)…”

Section: )mentioning

confidence: 99%

Weak universality of dynamical $$\Phi ^4_3$$ Φ 3 4 : non-Gaussian noise

Shen

2017

Stoch PDE: Anal Comp

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We consider a class of continuous phase coexistence models in three spatial dimensions. The fluctuations are driven by symmetric stationary random fields with sufficient integrability and mixing conditions, but not necessarily Gaussian. We show that, in the weakly nonlinear regime, if the external potential is a symmetric polynomial and a certain average of it exhibits pitchfork bifurcation, then these models all rescale to 4 3 near their critical point.

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Deterministic and stochastic hydrodynamic equations arising from simple microscopic model systems

Cited by 78 publications

References 0 publications

Introduction to Regularity Structures

Introduction to Regularity Structures

Diffusive Fluctuations for One-Dimensional Totally Asymmetric Interacting Random Dynamics

Weak universality of dynamical $$\Phi ^4_3$$ Φ 3 4 : non-Gaussian noise

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