KPZ equation, its renormalization and invariant measures

Abstract

The Kardar–Parisi–Zhang (KPZ) equation is a stochastic partial differential equation which is ill-posed because of the inconsistency between the nonlinearity and the roughness of the forcing noise. However, its Cole–Hopf solution, defined as the logarithm of the solution of the linear stochastic heat equation (SHE) with a multiplicative noise, is a mathematically well-defined object. In fact, Hairer (Ann Math 178:559–694, 2013) has recently proved that the solution of SHE can actually be derived through the Cole–Hopf transform of the solution of the KPZ equation with a suitable renormalization under periodic boundary conditions. This transformation is unfortunately not well adapted to studying the invariant measures of these Markov processes. The present paper introduces a different type of regularization for the KPZ equation on the whole line \({\mathbb {R}}\) or under periodic boundary conditions, which is appropriate from the viewpoint of studying the invariant measures. The Cole–Hopf transform applied to this equation leads to an SHE with a smeared noise having an extra complicated nonlinear term. Under time average and in the stationary regime, it is shown that this term can be replaced by a simple linear term, so that the limit equation is the linear SHE with an extra linear term with coefficient \(\tfrac{1}{24}\). The methods are essentially stochastic analytic: The Wiener–Itô expansion and a similar method for establishing the Boltzmann–Gibbs principle are used. As a result, it is shown that the distribution of a two-sided geometric Brownian motion with a height shift given by Lebesgue measure is invariant under the evolution determined by the SHE on \({\mathbb {R}}\).

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Acknowledgments

The authors thank Makiko Sasada for pointing out a simple proof of (3.32). T. Funaki was supported in part by the JSPS Grants (A) 22244007, (B) 26287014 and 26610019. J. Quastel was supported by Natural Sciences and Engineering Research Council of Canada, a Killam Fellowship, and the Institute for Advanced Study.

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Correspondence to Tadahisa Funaki.

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Funaki, T., Quastel, J. KPZ equation, its renormalization and invariant measures. Stoch PDE: Anal Comp 3, 159–220 (2015). https://doi.org/10.1007/s40072-015-0046-x

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Keywords

  • Invariant measure
  • Stochastic partial differential equation
  • KPZ equation
  • Cole–Hopf transform

Mathematics Subject Classification

  • 60H15
  • 82C28