A Rough Path to KPZ

This post is a collection of materials that are related to the KPZ equation. It is, in any measure, not complete. I just want to have a good repository of references on the subject. Also, it is a way for me to organize knowledge.

References

[Qua] Jeremy Quastel, Introduction to KPZ (the name says it all)
[HZ95] Timothy Halpin-Healy, Yi-Cheng Zhang, Kinetic roughening phenomena, stochastic growth, directed polymers and all that (explaining many models, good for motivation).

Introduction

Regularity Structures

The purpose of regularity structures, introduced in [Hai14] and motivated by [Lyo98, Gub04], is to generalise Taylor expansions using essentially arbitrary functions/distributions instead of polynomials.

Rough path results in regularity structures framework

Presentation

Eden model is not symmetric with respect to the origin as the cells develop in biases in one direction. The direction that has more cells has more probability to grow more cells.

So in Hairer’s theory, a function is called smooth if locally it can be approximated by the noise (and higher order terms constructed from the noise). This induces a natural topology in which the solutions to semilinear SPDEs depend continuously on the driving signal.

Outline:

  • Intro KPZ: IvanCorwin (Exactly SolveKPZ)/KPZ Universaltiy/IvanCorwin research statement
  • Zambotti in conjunction with Hairer-Friz
  • Brault $-$ $dy_t = F(y_t)dW_t$
  • Hairer Solving KPZ

Solving KPZ: the idea is to build a regularity structure that describes the KPZ, then using the reconstruction theorem to rebuild the distributions, which satisfy the normalized KPZ. Then show that it converge to the same limiting process independently of the choice of normalization.

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