Paracontrolled calculus for quasilinear singular PDEs

Bailleul, Ismaël; Mouzard, Antoine

doi:10.1007/s40072-022-00239-9

Cited by 2 publications

(1 citation statement)

References 32 publications

(24 reference statements)

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“…Some of these works are discussed in more detail in Section 1.3 below. Quasilinear SPDEs have been studied using regularity structures, rough path based methods and paracontrolled distribution theory, see [GH19b,OW19,BM22]. Recently, an approach inspired by the theory of regularity structures, but technically distinct, has been developed in [OSSW21,LOT21,LO22].…”

Section: Related Literaturementioning

confidence: 99%

Singular SPDEs on Homogeneous Lie Groups

Mayorcas¹,

Singh²

2023

Preprint

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The aim of this article is to extend the scope of the theory of regularity structures in order to deal with a large class of singular SPDEs of the formwhere the differential operator L fails to be elliptic. This is achieved by interpreting the base space R d as a non-trivial homogeneous Lie group G such that the differential operator ∂ t − L becomes a translation invariant hypoelliptic operator on G. Prime examples are the kinetic Fokker-Planck operator ∂ t − ∆ v − v • ∇ x and heat-type operators associated to sub-Laplacians. As an application of the developed framework, we solve a class of parabolic Anderson type equationson the compact quotient of an arbitrary Carnot group.

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Section: Related Literaturementioning

confidence: 99%

Singular SPDEs on Homogeneous Lie Groups

Mayorcas¹,

Singh²

2023

Preprint

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A diagram-free approach to the stochastic estimates in regularity structures

Linares,

Otto,

Tempelmayr

et al. 2024

Invent. math.

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In this paper, we explore the version of Hairer’s regularity structures based on a greedier index set than trees, as introduced in (Otto et al. in A priori bounds for quasi-linear SPDEs in the full sub-critical regime, 2021, arXiv:2103.11039) and algebraically characterized in (Linares et al. in Comm. Am. Math. Soc. 3:1–64, 2023). More precisely, we construct and stochastically estimate the renormalized model postulated in (Otto et al. in A priori bounds for quasi-linear SPDEs in the full sub-critical regime, 2021, arXiv:2103.11039), avoiding the use of Feynman diagrams but still in a fully automated, i. e. inductive way. This is carried out for a class of quasi-linear parabolic PDEs driven by noise in the full singular but renormalizable range. We assume a spectral gap inequality on the (not necessarily Gaussian) noise ensemble. The resulting control on the variance of the model naturally complements its vanishing expectation arising from the BPHZ-choice of renormalization. We capture the gain in regularity on the level of the Malliavin derivative of the model by describing it as a modelled distribution. Symmetry is an important guiding principle and built-in on the level of the renormalization Ansatz. Our approach is analytic and top-down rather than combinatorial and bottom-up.

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Paracontrolled calculus for quasilinear singular PDEs

Cited by 2 publications

References 32 publications

Singular SPDEs on Homogeneous Lie Groups

Singular SPDEs on Homogeneous Lie Groups

A diagram-free approach to the stochastic estimates in regularity structures

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