KPZ Reloaded

By Massimiliano Gubinelli, Nicolas Perkowski. 2015

224 citations recorded by the source corpus.

We analyze the one-dimensional periodic Kardar–Parisi–Zhang equation in the language of paracontrolled distributions, giving an alternative viewpoint on the seminal results of Hairer. Apart from deriving a basic existence and uniqueness result for paracontrolled solutions to the KPZ equation we perform a thorough study of some related problems. We rigorously prove the links between the KPZ equation, stochastic Burgers equation, and (linear) stochastic heat equation and also the existence of solutions starting from quite irregular initial conditions. We also show that there is a natural approximation scheme for the nonlinearity in the stochastic Burgers equation. Interpreting the KPZ equation as the value function of an optimal control problem, we give a pathwise proof for the global existence of solutions and thus for the strict positivity of solutions to the stochastic heat equation. Moreover, we study Sasamoto–Spohn type discretizations of the stochastic Burgers equation and show that their limit solves the continuous Burgers equation possibly with an additional linear transport term. As an application, we give a proof of the invariance of the white noise for the stochastic Burgers equation that does not rely on the Cole–Hopf transform.

Subfields: Finance, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/f4a6526f55ed89b7760348ac378968f0397f655c

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Citations in the corpus

A diagram-free approach to the stochastic estimates in regularity structures (2021, background): “Kupiainen appealed to Wilsonian renormalization [27, 28] to treat some semi-linear SPDEs.”
The Dynamic $${\Phi^4_3}$$Φ34 Model Comes Down from Infinity (2016, methodology): “Related results were obtained for the KPZ equation, first in [2] via a Cole–Hopf transformation, and then, following [22], in a series of works including [11,17,19,20,27,29].”
Discretisations of rough stochastic PDEs (2015, unknown): “ticle was nearing its completion, Zhu and Zhu [ZZ15] independently obtained the convergence of solutions to (Φ4 3,ε) to those of (Φ 4 3) using different methods. Additionally, Gubinelli and Perkowski [GP17] recently obtained a similar result for the KPZ equation. One advantage of the approach pursue”
HIGH ORDER PARACONTROLLED CALCULUS (2016, background): “While this kind of approach may seem far from being as powerful as Hairer’s machinery, the first order paracontrolled approach to singular stochastic PDEs has been successful in recovering and extending a number of results that can be proved within the setting of regularity structures, on the parabo”
PARACONTROLLED DISTRIBUTIONS AND SINGULAR PDES (2012, methodology): “The paracontrolled analysis of kpz will be presented elsewhere [GP14].”
Lectures on singular stochastic PDEs (2015, unknown): “These results will be included in [GP14].”