Local Solution to the Multi-layer KPZ Equation

  • Ajay Chandra
  • Dirk ErhardEmail author
  • Hao Shen


In this article we prove local well-posedness of the system of equations \(\partial _t h_{i}= \sum _{j=1}^{i}\partial _x^2 h_{j}+ (\partial _x h_{i})^2 + \xi \) on the circle where \(1\le i\le N\) and \(\xi \) is a space-time white noise. We attempt to generalize the renormalization procedure which gives the Hopf-Cole solution for the single layer equation and our \(h_1\) (solution to the first layer) coincides with this solution. However, we observe that cancellation of logarithmic divergences that occurs at the first layer does not hold at higher layers and develop explicit combinatorial formulae for them.


Renormalization Regularity structures Stochastic partial differential equations 



A. Chandra gratefully acknowledges financial support from the Leverhulme Trust via an Early Career Fellowship, ECF-2017-226. H. Shen gratefully acknowledges financial support by the NSF Award DMS-1712684 and DMS-1909525. A. Chandra and H. Shen would also like to thank the Isaac Newton Institute for Mathematical Science for support and hospitality during the programme “Scaling limits, rough paths, and quantum field theory”, supported by EPSRC Grant Number EP/R014604/1, where work on this paper was undertaken. D. Erhard gratefully acknowledges financial support from the National Council for Scientific and Technological Development – CNPq via a Universal grant.


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© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Imperial College LondonLondonUK
  2. 2.Universidade Federal da BahiaSalvadorBrazil
  3. 3.University of Wisconsin-MadisonMadisonUSA

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