Rough stochastic PDEs

Hairer, Martin

doi:10.1002/cpa.20383

Cited by 63 publications

(113 citation statements)

References 33 publications

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“…It is then possible to simply "postulate" the values of the integrals X s,t =: satisfying "Chen's relations" X s,t − X s,u − X u,t = X s,u ⊗ X u,t , (1.4) as well as the analytic bound |X s,t | |t − s| 2α , and to exploit this additional data to give a coherent definition of expressions of the type Y dX, provided that the path X is "enhanced" with its iterated integrals X and Y is a "controlled path" of the type (1.2). See for example [Gub04] for more information or [Hai11] for a concise exposition of this theory.…”

Section: Remark 12mentioning

confidence: 99%

Introduction to Regularity Structures

Friz

Hairer²

2014

Universitext

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These are short notes from a series of lectures given at the University of Rennes in June 2013, at the University of Bonn in July 2013, at the XVII th Brazilian School of Probability in Mambucaba in August 2013, and at ETH Zurich in September 2013. They give a concise overview of the theory of regularity structures as exposed in the article [Hai14]. In order to allow to focus on the conceptual aspects of the theory, many proofs are omitted and statements are simplified. We focus on applying the theory to the problem of giving a solution theory to the stochastic quantisation equations for the Euclidean Φ 4 3 quantum field theory.

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Section: Remark 12mentioning

confidence: 99%

Introduction to Regularity Structures

Friz

Hairer²

2014

Universitext

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“…The argument is essentially the same as the one given in [Hai11], which in turn relies very heavily on the results in [FV10a,FV10b], so we only explain the main steps and refer to [Hai11] for more details.…”

Section: Construction Of the Area Processmentioning

confidence: 99%

“…Special thanks are due to Hendrik Weber for numerous suggestions and his careful reading of the draft manuscript, as well as to Gérard Ben Arous who suggested that the techniques developed in [Hai12,Hai11] might prove useful for analysing the KPZ equation.…”

Section: Acknowledgementsmentioning

confidence: 99%

“…For more details, see the original article [Lyo98] and the monographs [LQ02, LCL07,FV10b] or, for a simplified exposition covering most of the notions required for this work, see [Hai11]. We will mostly make use of the notations and terminology introduced by Gubinelli in [Gub04] since the estimates given in that work seem to be the ones that are most suitable for the present undertaking.…”

Section: Elements Of Rough Path Theorymentioning

confidence: 99%

“…In all of these cases, the theory of rough paths was used to deal with the lack of temporal regularity of the equations. In this article, as in [Hai11,HW10], we use it instead in order to deal with the lack of spatial regularity.…”

Section: Treatment Of the Remaindermentioning

confidence: 99%

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Solving the KPZ Equation

Hairer

2013

XVIIth International Congress on Mathematical Physics

Self Cite

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We introduce a new concept of solution to the KPZ equation which is shown to extend the classical Cole-Hopf solution. This notion provides a factorisation of the Cole-Hopf solution map into a "universal" measurable map from the probability space into an explicitly described auxiliary metric space, composed with a new solution map that has very good continuity properties. The advantage of such a formulation is that it essentially provides a pathwise notion of a solution, together with a very detailed approximation theory. In particular, our construction completely bypasses the Cole-Hopf transform, thus laying the groundwork for proving that the KPZ equation describes the fluctuations of systems in the KPZ universality class.As a corollary of our construction, we obtain very detailed new regularity results about the solution, as well as its derivative with respect to the initial condition. Other byproducts of the proof include an explicit approximation to the stationary solution of the KPZ equation, a well-posedness result for the FokkerPlanck equation associated to a particle diffusing in a rough space-time dependent potential, and a new periodic homogenisation result for the heat equation with a space-time periodic potential. One ingredient in our construction is an example of a non-Gaussian rough path such that the area process of its natural approximations needs to be renormalised by a diverging term for the approximations to converge.

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Approximating Rough Stochastic PDEs

Hairer

Maas

Weber

2013

Comm Pure Appl Math

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We study approximations to a class of vector‐valued equations of Burgers type driven by a multiplicative space‐time white noise. A solution theory for this class of equations has been developed recently in Probability Theory Related Fields by Hairer and Weber. The key idea was to use the theory of controlled rough paths to give definitions of weak/mild solutions and to set up a Picard iteration argument. In this article the limiting behavior of a rather large class of (spatial) approximations to these equations is studied. These approximations are shown to converge and convergence rates are given, but the limit may depend on the particular choice of approximation. This effect is a spatial analogue to the Itô‐Stratonovich correction in the theory of stochastic ordinary differential equations, where it is well known that different approximation schemes may converge to different solutions.© 2014 Wiley Periodicals, Inc.

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Rough stochastic PDEs

Cited by 63 publications

References 33 publications

Introduction to Regularity Structures

Introduction to Regularity Structures

Solving the KPZ Equation

Approximating Rough Stochastic PDEs

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