Stochastic PDEs, Regularity Structures, and Interacting Particle Systems

Chandra, Ajay; Weber, H.

doi:10.48550/arxiv.1508.03616

Cited by 5 publications

(9 citation statements)

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“…Da Prato-Debussche approach. It is known since the breakthrough paper [dPD03] (see also the recent surveys on the topic [CW15,Hai14b,Hai15]) that in order to give a meaning (after suitable renormlisation) to such SPDEs, it is very convenient to make a first-order expansion around the solution of the linearised equation ∂ t Φ = 1 4π ∆Φ + ξ (the stochastic heat equation) as follows:…”

Section: The Case Of the Torus Tmentioning

confidence: 99%

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Dynamical Liouville

Garban

2020

Journal of Functional Analysis

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The aim of this paper is to analyze an SPDE which arises naturally in the context of Liouville quantum gravity. This SPDE shares some common features with the so-called Sine-Gordon equation and is built to preserve the Liouville measure which has been constructed recently on the two-dimensional sphere S 2 and the torus T 2 in the work by David-Kupiainen-Rhodes-Vargas [DK+16, DRV16]. The SPDE we shall focus on has the following (simplified) form:The main aspect which distinguishes this singular stochastic SPDE with well-known SPDEs studied recently (KPZ, dynamical Φ 4 3 , dynamical Sine-Gordon, generalized KPZ, etc.) is the presence of intermittency. One way of picturing this effect is that a naive rescaling argument suggests the above SPDE is sub-critical for all γ > 0, while we don't expect solutions to exist when γ > 2. In this work, we initiate the study of this intermittent SPDE by analyzing what one might call the "classical" or "Da Prato-Debussche" phase which corresponds here to γ ∈ [0, γ dP D = 2 √ 2 − √ 6). By exploiting the positivity of the non-linearity e γX , we can push this classical threshold further and obtain this way a weaker notion of solution when γ ∈ [γ dP D , γ pos = 2 √ 2 − 2). Our proof requires an analysis of the Besov regularity of natural space/time Gaussian multiplicative chaos (GMC) measures. Regularity Structures of arbitrary high degree should potentially give strong solutions all the way to the same threshold γ pos and should not push this threshold further unless the notion of regularity is suitably adapted to the present intermittent situation. Of independent interest, we prove along the way (using techniques from [HS16]) a stronger convergence result for approximate GMC measures µ → µ which holds in Besov spaces.

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Section: The Case Of the Torus Tmentioning

confidence: 99%

“…Following [Hai14a,CW15,Hai14b], we introduce the following space-time Besov spaces of negative regularity. Definition 2.1.…”

Section: The Case Of the Torus Tmentioning

confidence: 99%

Dynamical Liouville

Garban

2020

Journal of Functional Analysis

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“…Related results were obtained for the KPZ equation, first in [2] via a Cole-Hopf transformation, and then, following [22], in a series of works including [17,11,29,20,19,27]. See also the survey articles [24,6] for a summary of the work on the Φ 4 model with regularity structures. 1.3.…”

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confidence: 91%

The Dynamic $${\Phi^4_3}$$ Φ 3 4 Model Comes Down from Infinity

Mourrat

Weber

2017

Commun. Math. Phys.

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Abstract:We prove an a priori bound for the dynamic 4 3 model on the torus which is independent of the initial condition. In particular, this bound rules out the possibility of finite time blow-up of the solution. It also gives a uniform control over solutions at large times, and thus allows one to construct invariant measures via the Krylov-Bogoliubov method. It thereby provides a new dynamic construction of the Euclidean 4 3 field theory on finite volume. Our method is based on the local-in-time solution theory developed recently by Gubinelli, Imkeller, Perkowski and Catellier, Chouk. The argument relies entirely on deterministic PDE arguments (such as embeddings of Besov spaces and interpolation), which are combined to derive energy inequalities.

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“…We refer readers looking for a detailed exposition to [Hai15a], [FH14][Ch. 15], and [CW15]; our description of the theory will be quite brief. The most basic object in the theory is a regularity structure which consists of a triple (A, T , G).…”

Section: Regularity Structuresmentioning

confidence: 99%

Moment bounds for SPDEs with non-Gaussian fields and application to the Wong-Zakai problem

Chandra¹,

Shen²

2017

Electron. J. Probab.

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Upon its inception the theory of regularity structures [Hai14] allowed for the treatment for many semilinear perturbations of the stochastic heat equation driven by space-time white noise. When the driving noise is non-Gaussian the machinery of the theory can still be used but must be combined with an infinite number of stochastic estimates in order to compensate for the loss of hypercontractivity, as was done in [HS15]. In this paper we obtain a more streamlined and automatic set of criteria implying these estimates which facilitates the treatment of some other problems including non-Gaussian noise such as some general phase coexistence models [HX16], [SX16] -as an example we prove here a generalization of the Wong-Zakai Theorem found in [HP15].

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Stochastic PDEs, Regularity Structures, and Interacting Particle Systems

Cited by 5 publications

References 0 publications

Dynamical Liouville

Dynamical Liouville

The Dynamic $${\Phi^4_3}$$ Φ 3 4 Model Comes Down from Infinity

Moment bounds for SPDEs with non-Gaussian fields and application to the Wong-Zakai problem

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