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

Shen, Hao; Xu, Weijun

doi:10.1007/s40072-017-0107-4

Cited by 15 publications

(15 citation statements)

References 27 publications

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“…Similar results in higher dimensions d = 2, 3 were conjectured in [GLP99] but a complete proof in the two dimensional case was given only recently [MW16]. A similar convergence result is expected to hold in three dimensions, though a complete proof has not been established yet; however in [HX16,SX16] it was shown that a class of continuous phase coexistence models rescale to Φ 4 3 . 1 The tunable parameter in all of the results on convergence of variants of the asymmetric simple exclusion process to KPZ, is the asymmetry of the exclusion process: making it smaller and smaller corresponds to making the model locally more "Gaussian" which in turn corresponds to the fact that the dynamics on small scales are dominated by solutions of the linear equation.…”

Section: Introductionmentioning

confidence: 56%

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

Shen

Weber

2018

Journal of Functional Analysis

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We study the Glauber dynamics of a two dimensional Blume-Capel model (or dilute Ising model) with Kac potential parametrized by (β, θ) -the "inverse temperature" and the "chemical potential". We prove that the locally averaged spin field rescales to the solution of the dynamical Φ 4 equation near a curve in the (β, θ) plane and to the solution of the dynamical Φ 6 equation near one point on this curve. Our proof relies on a discrete implementation of Da Prato-Debussche method [DPD03] as in [MW16] but an additional coupling argument is needed to show convergence of the linearized dynamics.

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Section: Introductionmentioning

confidence: 56%

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

Shen

Weber

2018

Journal of Functional Analysis

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“…Moreover our construction is not restricted to the Gaussian setting and applies to any choice of the driving noise with minimal integrability conditions. In particular this allows to recover all the renormalisation procedures used so far in applications of the theory [32,[38][39][40]42,51]. It reaches however far beyond this and shows that the BPHZ renormalisation procedure belongs to the renormalisation group of the regularity structure associated to any class of subcritical semilinear stochastic PDEs.…”

Section: Mathematics Subject Classification 16t05 • 82c28 • 60h15 1 Imentioning

confidence: 76%

Algebraic renormalisation of regularity structures

2018

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We give a systematic description of a canonical renormalisation procedure of stochastic PDEs containing nonlinearities involving generalised functions. This theory is based on the construction of a new class of regularity structures which comes with an explicit and elegant description of a subgroup of their group of automorphisms. This subgroup is sufficiently large to be able to implement a version of the BPHZ renormalisation prescription in this context. This is in stark contrast to previous works where one considered regularity structures with a much smaller group of automorphisms, which lead to a much more indirect and convoluted construction of a renormalisation group acting on the corresponding space of admissible models by continuous transformations. Our construction is based on bialgebras of decorated coloured forests in cointeraction. More precisely, we have two Hopf algebras in cointeraction, coacting jointly on a vector space which represents the generalised functions of the theory. Two twisted antipodes play a fundamental role in the construc-B L. Zambotti

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“…Using the machinery developed there Hairer and Wu [11] proved a weak universality result for three dimensional reaction-diffusion equations in the case of Gaussian noise and a polynomial non-linearity, within the context of regularity structures. Weak universality for reaction-diffusion equations driven by non Gaussian noise is analysed in Shen and Wu [22]. Recently, important results concerning the stochastic quantisation equation we obtained by Mourrat and Weber.…”

Section: Theorem 11 (Convergence Of the Solution)mentioning

confidence: 99%

Weak universality for a class of 3d stochastic reaction–diffusion models

Furlan

Gubinelli

2018

Probab. Theory Relat. Fields

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We establish the large scale convergence of a class of stochastic weakly nonlinear reaction-diffusion models on a three dimensional periodic domain to the dynamic Φ 4 3 model within the framework of paracontrolled distributions. Our work extends previous results of Hairer and Xu to nonlinearities with a finite amount of smoothness (in particular C 9 is enough). We use the Malliavin calculus to perform a partial chaos expansion of the stochastic terms and control their L p norms in terms of the graphs of the standard Φ 4 3 stochastic terms.

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

Cited by 15 publications

References 27 publications

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

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

Algebraic renormalisation of regularity structures

Weak universality for a class of 3d stochastic reaction–diffusion models

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