Glauber dynamics of 2D Kac–Blume–Capel model and their stochastic PDE limits

Shen, Hao; Weber, Hendrik

doi:10.1016/j.jfa.2017.12.014

Cited by 15 publications

(13 citation statements)

References 40 publications

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“…The martingale noise dm is difficult to deal with since it depends nontrivially on h, and there are a fairly large number of such objects that need to be handled. Works such as [MW17a, SW18,Mat18] have made progress in studying convergence of approximate SPDEs driven by martingales, but a complete treatment for the KPZ is still under progress.…”

Section: Kpz Equation With General Nonlinearity Consider the Kpz Equmentioning

confidence: 99%

Some recent progress in singular stochastic partial differential equations

Corwin

Shen

2019

Bull. Amer. Math. Soc.

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Stochastic PDEs are ubiquitous in mathematical modeling. Yet, many such equations are too singular to admit classical treatment. In this article we review some recent progress in defining, approximating and studying the properties of a few examples of such equations. We focus mainly on the dynamical Φ 4 equation, KPZ equation and Parabolic Anderson Model, as well as touch on a few other equations which arise mainly in physics.

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Section: Kpz Equation With General Nonlinearity Consider the Kpz Equmentioning

confidence: 99%

Some recent progress in singular stochastic partial differential equations

Corwin

Shen

2019

Bull. Amer. Math. Soc.

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“…As in (2.6), the powers in the above SPDE have to be renormalised in order to find a nontrivial solution. The precise way the process is renormalised follows [MW17a,SW16]. Consider at first Z(t) the solution of the stochastic heat equation…”

Section: Solution Of the Limiting Equationmentioning

confidence: 99%

Tightness of the Ising–Kac Model on the Two-Dimensional Torus

Hairer

Iberti

2018

J Stat Phys

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We consider the sequence of Gibbs measures of Ising models with Kac interaction defined on a periodic two-dimensional discrete torus near criticality. Using the convergence of the Glauber dynamic proven by H. Weber and J.C. Mourrat [MW17a] and a method by H. Weber and P. Tsatsoulis employed in [TW16], we show tightness for the sequence of Gibbs measures of the Ising-Kac model near criticality and characterise the law of the limit as the Φ 4 2 measure on the torus. Our result is very similar to the one obtained by M. Cassandro, R. Marra, E. Presutti [CMP95] on Z 2 , but our strategy takes advantage of the dynamic, instead of correlation inequalities. In particular, our result covers the whole critical regime and does not require the large temperature / large mass / small coupling assumption present in earlier results.

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“…This argument has been applied to show the convergence of lattice systems to the KPZ equation [21], the Φ 4 3 equation [47], and to the parabolic Anderson model [10], and the most technical part of the proof was always the analysis of the random operator. The same argument was also applied to prove the convergence of the Kac-Ising / Kac-Blume-Capel model [37,42] to the Φ 4 2 / Φ 6 2 equation. This case can be handled without paracontrolled distributions, but also here some work is necessary to control the Fourier shuffle operator.…”

Section: Introductionmentioning

confidence: 99%

Paracontrolled distributions on Bravais lattices and weak universality of the 2d parabolic Anderson model

Martin¹,

Perkowski²

2019

Ann. Inst. H. Poincaré Probab. Statist.

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We develop a discrete version of paracontrolled distributions as a tool for deriving scaling limits of lattice systems, and we provide a formulation of paracontrolled distributions in weighted Besov spaces. Moreover, we develop a systematic martingale approach to control the moments of polynomials of i.i.d. random variables and to derive their scaling limits. As an application, we prove a weak universality result for the parabolic Anderson model: We study a nonlinear population model in a small random potential and show that under weak assumptions it scales to the linear parabolic Anderson model. MSC: 60H15, 60F05, 30H25

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Glauber dynamics of 2D Kac–Blume–Capel model and their stochastic PDE limits

Cited by 15 publications

References 40 publications

Some recent progress in singular stochastic partial differential equations

Some recent progress in singular stochastic partial differential equations

Tightness of the Ising–Kac Model on the Two-Dimensional Torus

Paracontrolled distributions on Bravais lattices and weak universality of the 2d parabolic Anderson model

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