Paracontrolled quasilinear SPDEs

Furlan, Marco; Gubinelli, Massimiliano

doi:10.1214/18-aop1280

Cited by 31 publications

(49 citation statements)

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“…The only disadvantage of our approach, compared to [1,5,11], is that it is not obvious at all a priori why the counterterms generated by the renormalization procedure should be local in the solution. The reason for this is that our method relies on the introduction of additional "nonphysical" components to our equation, which are given by some nonlocal, nonlinear functionals of the solution, and we cannot rule out in general that the counterterms depend on these nonphysical terms.…”

Section: Introductionmentioning

confidence: 99%

“…x > 0, a is a smooth function taking values in K for some compact K .0; 1/, and F 0 and F 1 are smooth functions. The quasilinear equations considered in previous works [1,5,11] correspond to situations where > 1 3 , F 0 D 0. Let us take a compactly supported, nonnegative, symmetric (under the involution x 7 !…”

Section: Introductionmentioning

confidence: 99%

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confidence: 99%

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A Solution Theory for Quasilinear Singular SPDEs

Gerencsér

Hairer

2019

Comm Pure Appl Math

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We give a construction allowing us to build local renormalized solutions to general quasilinear stochastic PDEs within the theory of regularity structures, thus greatly generalizing the recent results of [1, 5, 11]. Loosely speaking, our construction covers quasilinear variants of all classes of equations for which the general construction of [3, 4, 7] applies, including in particular one‐dimensional systems with KPZ‐type nonlinearities driven by space‐time white noise. In a less singular and more specific case, we furthermore show that the counterterms introduced by the renormalization procedure are given by local functionals of the solution. The main feature of our construction is that it allows exploitation of a number of existing results developed for the semilinear case, so that the number of additional arguments it requires is relatively small. © 2018 the Authors. Communications on Pure and Applied Mathematics is published by the Courant Institute of Mathematical Sciences and Wiley Periodicals, Inc.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A Solution Theory for Quasilinear Singular SPDEs

Gerencsér

Hairer

2019

Comm Pure Appl Math

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“…These two theories allowed for the first time to study stochastic PDEs with very singular coefficients (such as the Kardar-Parisi-Zhang equation, see [15]) which posed long standing problems. Amongst the many papers in the area of stochastic PDEs that build on these ideas, we mention a series of recent ones on quasilinear stochastic PDEs [2,10,13,24,25] that may be of interest to the reader. Also in the present paper we consider a quasilinear PDE, but a deterministic one, where one of the coefficients is singular because it is a distribution.…”

Section: Introductionmentioning

confidence: 99%

A non-linear parabolic PDE with a distributional coefficient and its applications to stochastic analysis

Issoglio

2019

Journal of Differential Equations

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We consider a non-linear parabolic partial differential equation (PDE) on R d with a distributional coefficient in the non-linear term. The distribution is an element of a Besov space with negative regularity and the non-linearity is of quadratic type in the gradient of the unknown. Under suitable conditions on the parameters we prove local existence and uniqueness of a mild solution to the PDE, and investigate properties like continuity with respect to the initial condition and blowup times. We prove a global existence and uniqueness result assuming further properties on the non-linearity. To conclude we consider an application of the PDE to stochastic analysis, in particular to a class of non-linear backward stochastic differential equations with distributional drivers.

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“…So a better understanding for more regular noises should be helpful. To deal with rough noises an alternative to regularity structures is to stay closer to a paracontrolled approach [31], which has recently been proposed for certain quasilinear SPDEs [8,27,50,51].Since the linear operator A depends on the solution itself, which will be in our case a stochastic process, we cannot apply the standard fixed-point argument as in [3,63]. Namely, if we denote with U u the random evolution operator generated by A(u), one naturally expects that the mild solution of (1.1) should be given by the variation-of-constants formula u(t) = U u (t, 0)u 0 + t 0 U u (t, s)F (s, u(s)) ds +t 0 U u (t, s)σ(s, u(s)) dW (s).(1.2)As already observed in [57], and justified in Sections 2 and 3, the random evolution operator…”

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Pathwise mild solutions for quasilinear stochastic partial differential equations

Kuehn

Neamţu

2020

Journal of Differential Equations

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Stochastic partial differential equations (SPDEs) have become a key modelling tool in applications. Yet, there are many classes of SPDEs, where the existence and regularity theory for solutions is not completely developed. Here we contribute to this aspect and prove the existence of mild solutions for a broad class of quasilinear Cauchy problems, includingamong others -cross-diffusion systems as a key application. Our solutions are local-in-time and are derived via a fixed point argument in suitable function spaces. The key idea is to combine the theory of deterministic quasilinear parabolic partial differential equations (PDEs) with recent theory of evolution semigroups. We also show, how to apply our theory to the Shigesada-Kawasaki-Teramoto (SKT) model. Furthermore, we provide examples of blow-up and ill-posed operators, which can occur after finite-time.Here we aim to develop a theory for quasilinear stochastic evolution equations (1.1) using a semigroup approach. The main theme is to extend the very general deterministic theory of quasilinear Cauchy problems [3,61,63]. The key idea is to employ a modified definition of mild solutions [57] for the quasilinear case in comparison to the more classical parabolic SPDE setting [54]. Before we describe our approach in more detail, we briefly review some other techniques and solution concepts used for (certain subclasses of) the SPDE (1.1). Instead of mild solutions, one may instead use weak, or martingale, solutions [13,20,24,35] of (1.1); here weak solution is interpreted in the classical PDE sense while these solutions are also sometimes referred to as strong solutions from a probabilistic perspective [55]. There are also several works exploiting the additional assumption of monotone coefficients [43] particularly in the case of the stochastic porous medium equation [9,10], where A(u) = ∆(a(u)) for a maximal monotone map a and F ≡ 0. Other approaches to quasilinear SPDEs are based upon a gradient structure [29], approximation methods [38,39], kinetic solutions [20,30], or directly looking at strong (in the PDE sense) solutions [36].One may ask, why one might want to prove the existence of pathwise mild solutions obtained by a suitable variations-of-constants/Duhamel formula [34] instead of working with weak solutions obtained in a formulation via test functions [26]? One reason is that a mild formulation is often more natural to work with in the context of (random) dynamical systems for SPDEs [18,34]. In fact, many classical results regarding dynamics and long-time behavior of semilinear SPDEs are often crucially based upon the mild formulation and semigroups [34]. We expect this theory to generalize a lot easier also in the quasilinear case if one does not have to work with weak(er) solutions. If we take the SKT system again as a motivation, then there are deterministic results regarding the existence of attractors using weak [53] as well as mild [62] solutions concepts. We intend to investigate the existence of random attractors for the stochastic SKT equation ...

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Paracontrolled quasilinear SPDEs

Abstract: We introduce a non-linear paracontrolled calculus and use it to renormalise a class of singular SPDEs including certain quasilinear variants of the periodic two dimensional parabolic Anderson model.

Cited by 31 publications

References 35 publications

A Solution Theory for Quasilinear Singular SPDEs

A Solution Theory for Quasilinear Singular SPDEs

A non-linear parabolic PDE with a distributional coefficient and its applications to stochastic analysis

Pathwise mild solutions for quasilinear stochastic partial differential equations

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