PARACONTROLLED DISTRIBUTIONS AND SINGULAR PDES

By Massimiliano Gubinelli, Peter Imkeller, Nicolas Perkowski. Forum of Mathematics, Pi, 2012

692 citations recorded by the source corpus.

We introduce an approach to study certain singular partial differential equations (PDEs) which is based on techniques from paradifferential calculus and on ideas from the theory of controlled rough paths. We illustrate its applicability on some model problems such as differential equations driven by fractional Brownian motion, a fractional Burgers-type stochastic PDE (SPDE) driven by space-time white noise, and a nonlinear version of the parabolic Anderson model with a white noise potential.

Subfields: Finance, Mathematical Physics, Modeling and Simulation

View Paper: https://www.semanticscholar.org/paper/7b03526fbe7a203ecf2a92ee3dc991d92c89031a

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Finance

In Mathematical Physics

In Modeling and Simulation

Citations in the corpus

Multiplicative stochastic heat equations on the whole space (2015, result): “m the equation. When the space variable is restricted to a torus of dimension 2, the solution of a generalised version of (PAM) has been constructed independently by Gubinelli, Imkeller and Perkowski [GIP15] using paracontrolled distributions, and by Hairer [Hai14b] via the theory of regularity stru”
Energy solutions of KPZ are unique (2015, methodology): “This difficulty persists also in other recent (but, in this case, mostly equivalent) methods of giving a meaning to the SBE: Hairer’s regularity structures [8] or Gubinelli, Imkeller and Perkowski’s paracontrolled distributions [6, 5].”
A priori estimates for rough PDEs with application to rough conservation laws (2016, background): “A more conservative approach, useful in many situations but not as general, is the paracontrolled calculus developed by Gubinelli, Imkeller and Perkowski [31,32].”
Stochastic heat equations with general multiplicative Gaussian noises: Hölder continuity and intermittency (2014, methodology): “In this context, definitions of the equation by means of renormalization and rough paths techniques have been recently investigated in [21, 23].”
The Dynamic $${\Phi^4_3}$$Φ34 Model Comes Down from Infinity (2016, methodology): “The analysis presented in this article is based on the paracontrolled approach of [5,18].”
Heat semigroup and singular PDEs (2015, methodology): “Following the recent breakthrough of Hairer [25] and Gubinelli, Imkeller, Perkowski [22], there has been recently a tremendous activity in the study of parabolic singular Date: January 28, 2015.”
A diagram-free approach to the stochastic estimates in regularity structures (2021, methodology): “Gubinelli & Imkeller & Perkowski’s paracontrolled calculus [18] provides an alternative approach, based on Littlewood-Paley decomposition, to (stochastic) estimates and renormalization.”
Renormalising SPDEs in regularity structures (2017, methodology): “frameworks available that provide rigorous descriptions of what it means to be a (local) solution to these SPDEs: the theory of regularity structures [Hai14], the theory of paracontrolleddistributions[GIP15],aWilsonianrenormalisationgroupapproach[Kup16], and most recently the approach of [OW19]. Alt”
Paracontrolled approach to the three-dimensional stochastic nonlinear wave equation with quadratic nonlinearity (2018, methodology): “See Section 2 and [3, 34] for further details.”
The infinitesimal generator of the stochastic Burgers equation (2018, background): “In recent years, new solution theories like regularity structures [20,40] or paracontrolled distributions [26,33] were developed for singular SPDEs, see [38] for an up-to-date and fairly exhaustive review.”
Paracontrolled distributions and the 3-dimensional stochastic quantization equation (2013, methodology): “More precisely we will expand the solution as the sum of stochastic objects involving the Gaussian field X and derive an equation for the remainder term, which can be solved by a fixed point argument using the notion of paracontrolled distributions introduced in [9].”
Renormalization Group and Stochastic PDEs (2016, background): “The latter case was also addressed by Catellier and Chouk [2] based on the theory of paracontrolled distributions developed in [3].”
Restricted Markov uniqueness for the stochastic quantization of $P(\Phi)_2$ and its applications (2015, methodology): “has arisen a renewed interest in SPDEs related to such problems, particularly in connection with Hairer’s theory of regularity structures [Hai14] and related work by Imkeller, Gubinelli, Perkowski in [GIP13]. By using these theories one can obtain local existence and uniqueness of solution to (1.1) ”
Discretisations of rough stochastic PDEs (2015, methodology): “In three dimensions, alternative approaches to ( (4)3) were recently obtained in [6] (via paracontrolled distributions, see [14] for the development of that approach), and in [30] (via renormalisation group techniques à la Wilson).”
HIGH ORDER PARACONTROLLED CALCULUS (2016, background): “See [20, 3] or [32] for a sample of works where u0 P Cα, in a first order paracontrolled setting.”