On hydrodynamic limits in Sinai-type random environments

Landim, Cláudio; Pacheco, Carlos G.; Sethuraman, Sunder; Xue, Jianfei

doi:10.48550/arxiv.2006.00583

Cited by 4 publications

(5 citation statements)

References 30 publications

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“…Viewing the solution Π to (19) as a function of ξ next to x and z, and writing z := (λ 1−α z x , λ α z 1 , λ 2α z 2 , . .…”

Section: Model Space and Structure Groupmentioning

confidence: 99%

“…The regime α > 1 corresponds to spatially colored noise, which has been studied in the articles [17] and [18]. We also mention the articles [6], [7], and [19] where singular quasi-linear SPDE's arise naturally in some relevant physical models.…”

Section: Introductionmentioning

confidence: 99%

“…and hence to interpret solutions to (19) as being functions of the variable z = (z x , z 1 , z 2 , . .…”

Section: Introductionmentioning

confidence: 99%

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A priori bounds for quasi-linear SPDEs in the full sub-critical regime

Otto¹,

Sauer²,

Smith³

et al. 2021

Preprint

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This paper is concerned with quasi-linear parabolic equations driven by an additive forcing ξ ∈ C α−2 , in the full subcritical regime α ∈ (0, 1). We are inspired by Hairer's regularity structures, however we work with a more parsimonious model indexed by multi-indices rather than trees. This allows us to capture additional symmetries which play a crucial role in our analysis. Assuming bounds on this model, which is modified in agreement with the concept of algebraic renormalization, we prove local a priori estimates on solutions to the quasi-linear equations modified by the corresponding counter terms.1 A number of aspects of this paper also work for arbitrary α > 0, but the authors did not identify the renormalized PDE in the full sub-critical regime.2 Extending the linear theory developed in [23] to arbitrary α > 0 remains an interesting and challenging open problem.

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“…Viewing the solution Π to (19) as a function of ξ next to x and z, and writing z := (λ 1−α z x , λ α z 1 , λ 2α z 2 , . .…”

Section: Model Space and Structure Groupmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

A priori bounds for quasi-linear SPDEs in the full sub-critical regime

Otto¹,

Sauer²,

Smith³

et al. 2021

Preprint

View full text Add to dashboard Cite

show abstract

“…Due to the discussions in Subsection 1.4, especially by Definition 1.1, the result for (1.4) implies that for ∇u for the solution u of (1.1). Our assumption a = g ′ for (1.1) corresponds to χ = ϕ for (1.4) so that we consider the equation of the special form (1.6) ∂ t v = ∆{ϕ(v)} + ∇{ϕ(v)ξ}, on T. The equation (1.6) has a physical meaning in the sense that it can be derived from a microscopic particle system in random environment, see [14].…”

Section: The Aim Of the Articlementioning

confidence: 99%

Global solvability and convergence to stationary solutions in singular quasilinear stochastic PDEs

Funaki¹,

Xie²

2021

Preprint

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We consider singular quasilinear stochastic partial differential equations (SPDEs) studied in [7], which are defined in paracontrolled sense. The main aim of the present article is to establish the global-in-time solvability for a particular class of SPDEs with origin in particle systems and, under a certain additional condition on the noise, prove the convergence of the solutions to stationary solutions as t → ∞. We apply the method of energy inequality and Poincaré inequality. It is essential that the Poincaré constant can be taken uniformly in an approximating sequence of the noise. We also use the continuity of the solutions in the enhanced noise, initial values and coefficients of the equation, which we prove in this article for general SPDEs discussed in [7] except that in the enhanced noise. Moreover, we apply the initial layer property of improving regularity of the solutions in a short time.

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“…In particular, for such empirical density fields progress has been recently made in the understanding of equilibrium and non-equilibrium fluctuations, including boundary dynamics or random environments, as well as considering more general geometries (see e.g. [18,13,6,17,21] and references therein).…”

Section: Introductionmentioning

confidence: 99%

Higher order hydrodynamics and equilibrium fluctuations of interacting particle systems

Chen,

2020

Preprint

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Motivated by the recent preprint [arXiv:2004.08412] by Ayala, Carinci, and Redig, we first provide a general framework for the study of scaling limits of higher order fields. Then, by considering the same class of infinite interacting particle systems as in [arXiv:2004.08412], namely symmetric simple exclusion and inclusion processes in the d-dimensional Euclidean lattice, we prove the hydrodynamic limit, and convergence for the equilibrium fluctuations, of higher order fields. In particular, the limit fields exhibit a tensor structure. Our fluctuation result differs from that in [arXiv:2004.08412], since we consider a different notion of higher order fluctuation fields.

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On hydrodynamic limits in Sinai-type random environments

Cited by 4 publications

References 30 publications

A priori bounds for quasi-linear SPDEs in the full sub-critical regime

A priori bounds for quasi-linear SPDEs in the full sub-critical regime

Global solvability and convergence to stationary solutions in singular quasilinear stochastic PDEs

Higher order hydrodynamics and equilibrium fluctuations of interacting particle systems

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