Solving the KPZ equation

By Martin Hairer. 2011

688 citations recorded by the source corpus.

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 uctuations 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 Fokker-Planck equation associated to a particle diusing 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.

Subfields: Computational Theory and Mathematics, Condensed Matter Physics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/612da56c76a474cd82c81150accf3ed8db36b1f6

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Citations in the corpus

Volatility estimation for stochastic PDEs using high-frequency observations (2017, background): “While in probability theory there are recently enormous efforts to advance research for SPDEs, for instance Hairer (2013) was able to solve the KPZ equation, there is scant groundwork on statistical inference for SPDEs and there remain many open questions.”
Lectures on singular stochastic PDEs (2015, methodology): “At the same time, Hairer managed to devise a vast generalization of the basic construction of controlled rough paths in the multidimensional and distributional setting, which he called the theory of regularity structures [Hai14] and which subsumes standard analysis based on Hölder spaces and control”
Anomalous Diffusion: A Basic Mechanism for the Evolution of Inhomogeneous Systems (2019, background): “However, the apparent simplicity of this equation is misleading, since the nonlinear gradient term combined with the noise makes it one of the toughest problems in modern mathematical physics [134, 135].”
Introduction to KPZ (2011, methodology): “3Technically, the results of [67] are for KPZ on [0, 2π) with periodic boundary conditions, and extending them to R remains an open problem.”
Lectures on integrable probability (2012, background): “ of understanding the growth of surfaces we started with(i.e.ballisticdeposition),see[Corwin-11]foranicerecentsurvey. Duetonon-linearityof (1.6)itistrickyeventogiveameaningtothisequation(see, however,[Hairer-11]forarecentprogress),butphysicistsstilldealtwithitandthat’sone wayhowtheexponent1=3 oft13 ”
Nonlinear Fluctuations of Weakly Asymmetric Interacting Particle Systems (2013, methodology): “Hairer [26] has provided a more satisfactory answer to this question.”
Nonlinear Fluctuating Hydrodynamics for Anharmonic Chains (2013, background): “In [49] a more sophisticated, and more general, RG construction is presented.”
Multiplicative stochastic heat equations on the whole space (2015, methodology): “ve the convergence of the ﬂuctuations oftheweaklyasymmetric simple exclusion process totheKPZequation onR. A more direct interpretation of the KPZ equation itself has recently been obtained by Hairer [Hai13], when the space variable is restricted to a torus of dimension 1. Inaddition to the ill-deﬁn”
A CLASS OF GROWTH MODELS RESCALING TO KPZ (2015, methodology): “ polynomials of arbitrary degree are allowed in the right hand side of (1.18), the number of objects that need to be explicitly controlled in the limit ε→ 0 can be very large. In the original article [Hai13], almost half of the article was devoted to the control of only ﬁve such obje cts. This was s”
Heat semigroup and singular PDEs (2015, methodology): “Hairer’s theory of regularity structures is built on the insights of earlier works [26, 29, 27] on (1+1)-dimensional space-time problems where he used the framework of rough paths theory, under the form of Gubinelli’s controlled paths, to make sense of previously ill-posed singular PDEs and give a m”
Discretisations of rough stochastic PDEs (2015, methodology): “This lays the foundations of a systematic approximation theory which can in principle be applied to many other singular SPDEs, for example, stochastic Burgers-type equations [17, 22, 23], the KPZ equation [3, 18, 29] or the continuous parabolic Anderson model [19, 21].”
A regularity structure for rough volatility (2017, methodology): “Example 3.6), then the renormalization 𝒞𝜀 = 𝒦𝜀(⋅, ⋅) that was applied in Theorem 3.14 and thus in Definition 3.24 is a constant, which is the familiar concept one encounters in the study of singular SPDE (Chandra & Hairer, 2016; Hairer, 2013, 2014).”
Height Fluctuations for the Stationary KPZ Equation (2014, background): “The physically relevant notion [3, 14, 33, 39, 56, 71, 74] of a solution to the KPZ equation is therefore defined indirectly via the well-posed stochastic heat equation (SHE) with multiplicative noise, ∂TZ(T,X) = 12∂ 2 XZ(T,X) + Z(T,X)ξ(T,X) with initial condition Z(0, X) = Z0(X) = eH(0,X).”
Rough differential equations with unbounded drift term (2016, methodology): “xpose the theory and its various applications. Let us also brieﬂy mention that rough paths ideas were used by M. Hairer to solve stochastic partial diﬀerential equations (SPDE) like the KPZ-equation ([Hai13]) and form an important part in his 2010 Mathematics Subject Classiﬁcation. 34A34, 34F05, 60G”
An appetizer to modern developments on the Kardar–Parisi–Zhang universality class (2017, methodology): “12 As an alternative approach, Hairer extended what is called the rough path theory in mathematics and successfully provided a mathematical foundation to the KPZ equation [47].”
HIGH ORDER PARACONTROLLED CALCULUS (2016, background): “in Lyons’ theory of controlled diﬀerential equations is universal, in the sense that it depends only on the dimension and the irregularity exponent of the control, the ground-breaking works of Hairer [16, 17] and Gubinelli-Imkeller-Perkowski [13] uncovered the fact that this space of enhanced noises”