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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.79.

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Energy solutions of KPZ are unique
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by Massimiliano Gubinelli and Nicolas Perkowski HTML | PDF
J. Amer. Math. Soc. 31 (2018), 427-471 Request permission

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
  • MR Author ID: 692441
  • Email: gubinelli@iam.uni-bonn.de
  • Nicolas Perkowski
  • Affiliation: Institut für Mathematik, Humboldt–Universität zu Berlin, Berlin, Germany
  • MR Author ID: 999469
  • Email: perkowsk@math.hu-berlin.de
  • 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.
  • © Copyright 2017 American Mathematical Society
  • Journal: J. Amer. Math. Soc. 31 (2018), 427-471
  • MSC (2010): Primary 60H15
  • DOI: https://doi.org/10.1090/jams/889
  • MathSciNet review: 3758149