# Local Solution to the Multi-layer KPZ Equation

- 20 Downloads

## Abstract

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.

## Keywords

Renormalization Regularity structures Stochastic partial differential equations## Notes

### Acknowledgements

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.

## References

- 1.Bruned, Y., Hairer, M., Zambotti, L.: Algebraic renormalisation of regularity structures (2016). arXiv preprint arXiv:1610.08468
- 2.Carlitz, L.: The product of several Hermite or Laguerre polynomials. Monatshefte für Mathematik
**66**(5), 393–396 (1962)MathSciNetCrossRefzbMATHGoogle Scholar - 3.Chandra, A., Hairer, M.: An analytic BPHZ theorem for regularity structures (2016). arXiv preprint arXiv:1612.08138
- 4.Funaki, T., Hoshino, M.: A coupled KPZ equation, its two types of approximations and existence of global solutions. J. Funct. Anal.
**273**(3), 1165–1204 (2017). https://doi.org/10.1016/j.jfa.2017.05.002 MathSciNetCrossRefzbMATHGoogle Scholar - 5.Hairer, M.: Solving the KPZ equation. Ann. Math. (2)
**178**(2), 559–664 (2013). https://doi.org/10.4007/annals.2013.178.2.4 MathSciNetCrossRefzbMATHGoogle Scholar - 6.Hairer, M.: A theory of regularity structures. Invent. Math.
**198**(2), 269–504 (2014). https://doi.org/10.1007/s00222-014-0505-4 ADSMathSciNetCrossRefzbMATHGoogle Scholar - 7.Hairer, M.: The motion of a random string (2016). arXiv preprint arXiv:1605.02192
- 8.Hairer, M., Quastel, J.: A class of growth models rescaling to KPZ. Forum Math. Pi
**6**, e3, 112 (2018). https://doi.org/10.1017/fmp.2018.2 - 9.Hairer, M., Shen, H.: A central limit theorem for the KPZ equation. Ann. Probab.
**45**(6B), 4167–4221 (2017)MathSciNetCrossRefzbMATHGoogle Scholar - 10.O’Connell, N., Warren, J.: A multi-layer extension of the stochastic heat equation. Commun. Math. Phys.
**341**(1), 1–33 (2016). https://doi.org/10.1007/s00220-015-2541-3 ADSMathSciNetCrossRefzbMATHGoogle Scholar