Local solution to the multi-layer KPZ equation

TL;DR

This paper establishes local well-posedness for a multi-layer KPZ system driven by space-time white noise, extending the renormalization approach used for the single-layer case and addressing divergence issues at higher layers.

Contribution

It generalizes the renormalization procedure for the multi-layer KPZ equations and provides explicit formulas for divergence cancellations at higher layers.

Findings

Proves local well-posedness of multi-layer KPZ system.

Extends renormalization techniques from single to multi-layer cases.

Develops explicit combinatorial formulas for divergence cancellations.

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_{i}+ (\partial_x h_{i})^2 + \xi $ on the circle where $1\leq i\leq N$ 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.

Equations

$${\partial }_t{h}_i=\sum _{j=1}^i{\partial }_x^2{h}_j+({\partial }_x{h}_i{)}^2+\xi i\in \{1,2,\cdots ,N\},(1):MKPZintro=e01$$

Full text

1. 1 Introduction

1 Introduction

The aim of this paper is to introduce a system of equations called the multi-layer KPZ equation, formally given by

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