Local bounds for stochastic reaction diffusion equations

Moinat, Augustin; Weber, Hendrik

doi:10.1214/19-ejp397

Cited by 9 publications

(21 citation statements)

References 24 publications

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“…This proof is a version of the proof of theorem 4.4 in our companion paper, [18], specialised to cubic nonlinearity. This specialisation makes the proof significantly simpler.…”

Section: Proof Of Lemma 27mentioning

confidence: 92%

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Space‐Time Localisation for the Dynamic Model

Moinat

Weber

2020

Comm Pure Appl Math

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We prove an a priori bound for solutions of the dynamic 4 3 equation. This bound provides a control on solutions on a compact space-time set only in terms of the realisation of the noise on an enlargement of this set, and it does not depend on any choice of space-time boundary conditions. We treat the large-and small-scale behaviour of solutions with completely different arguments. For small scales we use bounds akin to those presented in Hairer's theory of regularity structures. We stress immediately that our proof is fully self-contained, but we give a detailed explanation of how our arguments relate to Hairer's. For large scales we use a PDE argument based on the maximum principle. Both regimes are connected by a solution-dependent regularisation procedure. The fact that our bounds do not depend on space-time boundary conditions makes them useful for the analysis of large-scale properties of solutions. They can, for example, be used in a compactness argument to construct solutions on the full space and their invariant measures. © 2020 The Authors. Communications on Pure and Applied Mathematics published by Wiley Periodicals LLC set that depends only on a finite number of explicit polynomials in the Gaussian noise on a slightly larger space-time set. In particular, our bound does not depend on any space-time boundary conditions. The main difficulty when working with (1.1) is the roughness of the driving noise , which in turn makes the solution irregular and the interpretation of nonlinear terms nontrivial. It is now well-understood that solutions are distribution

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“…This proof is a version of the proof of theorem 4.4 in our companion paper, [18], specialised to cubic nonlinearity. This specialisation makes the proof significantly simpler.…”

Section: Proof Of Lemma 27mentioning

confidence: 92%

“…Finally, we would like to mention that in our companion paper [18] we have implemented our approach in the case of one-dimensional reaction-diffusion equations. Even in this much more regular case where no renormalisation enters, a priori bounds that do not depend on space-time boundary conditions seem to have been unknown.…”

Section: G ;mentioning

confidence: 99%

Space‐Time Localisation for the Dynamic Model

Moinat

Weber

2020

Comm Pure Appl Math

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“…A key ingredient for this proof is a powerful a priori bound that establishes a "coming down from infinity" property for ( 43 ). This kind of bound was first established via paracontrolled techniques in [27] and later a much shorter argument that is in flavour based on the theory of regularity structures was given in [28,29].…”

Section: Introductionmentioning

confidence: 98%

“…Our proof strongly relies on slight modifications of the a priori bounds obtained in [28], but is otherwise very elementary. As proposed by Parisi and Wu [32], we interpret μ as the invariant measure of the 4 3 equation [21], which was shown to exist in [27] and is unique by [24,25].…”

Section: Introductionmentioning

confidence: 99%

“…Both Gubinelli-Hofmanová [16] and Moinat-Weber [28] had previously obtained stretched exponential integrability for any exponent strictly less than 1 using SPDE techniques. Whilst this is sufficient to verify the regularity axiom in the form of [9], it is insufficient for the form stated in [19] for the purpose of simplifying the exposition there.…”

Section: Introductionmentioning

confidence: 99%

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The $$\Phi _3^4$$ Measure Has Sub-Gaussian Tails

Hairer

Steele

2022

J Stat Phys

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We provide a very simple argument showing that the $$\Phi ^4_3$$ Φ 3 4 measure does have quartic exponential tails, as expected from its formal expression. This shows that the corresponding moment problem is well-posed and provides a simple path to observing non-Gaussianity of the measure.

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Stochastic Quantization of an Abelian Gauge Theory

Shen

2021

Commun. Math. Phys.

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We study the Langevin dynamics of a U (1) lattice gauge theory on the torus, and prove that they converge for short time in a suitable gauge to a system of stochastic PDEs driven by space-time white noises. This also yields convergence of some gauge invariant observables on a short time interval. We fix gauge via a DeTurck trick, and prove a version of Ward identity which results in cancellation of renormalization constants that would otherwise break gauge symmetry. The proof relies on a discrete version of the theory of regularity structures.

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Local bounds for stochastic reaction diffusion equations

Cited by 9 publications

References 24 publications

Space‐Time Localisation for the Dynamic Model

Space‐Time Localisation for the Dynamic Model

The $$\Phi _3^4$$ Measure Has Sub-Gaussian Tails

Stochastic Quantization of an Abelian Gauge Theory

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