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Energy solutions of KPZ are unique


Authors: Massimiliano Gubinelli and Nicolas Perkowski
Journal: J. Amer. Math. Soc. 31 (2018), 427-471
MSC (2010): Primary 60H15
DOI: https://doi.org/10.1090/jams/889
Published electronically: October 19, 2017
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Abstract: The Kardar-Parisi-Zhang (KPZ) equation is conjectured to universally describe the fluctuations of weakly asymmetric interface growth. Here we provide the first intrinsic well-posedness result for the stationary KPZ equation on the real line by showing that its energy solutions, as introduced by Gonçalves and Jara in 2010 and refined by Gubinelli and Jara, are unique. This is the first time that a singular stochastic PDE can be tackled using probabilistic methods, and the combination of the convergence results of the first work and many follow-up papers with our uniqueness proof establishes the weak KPZ universality conjecture for a wide class of models. Our proof builds on an observation of Funaki and Quastel from 2015, and a remarkable consequence is that the energy solution to the KPZ equation is not equal to the Cole-Hopf solution, but it involves an additional drift $ t/12$.


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Additional Information

Massimiliano Gubinelli
Affiliation: Hausdorff Center for Mathematics & Institute for Applied Mathematics, Universität Bonn, Bonn, Germany
Email: gubinelli@iam.uni-bonn.de

Nicolas Perkowski
Affiliation: Institut für Mathematik, Humboldt–Universität zu Berlin, Berlin, Germany
Email: perkowsk@math.hu-berlin.de

DOI: https://doi.org/10.1090/jams/889
Received by editor(s): November 12, 2016
Received by editor(s) in revised form: July 30, 2017
Published electronically: October 19, 2017
Additional Notes: The first author gratefully acknowledges financial support by the DFG via CRC 1060.
The second author gratefully acknowledges financial support by the DFG via Research Unit FOR 2402.
Article copyright: © Copyright 2017 American Mathematical Society

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