Renormalized-Generalized Solutions for the KPZ Equation

TL;DR

This paper introduces a new solution concept for the KPZ equation that includes the Cole-Hopf solution, offering a pathwise framework and structured approximation based on distribution regularization.

Contribution

It develops a novel, pathwise solution framework for the KPZ equation that generalizes previous approaches and incorporates the Cole-Hopf solution within a distribution theory context.

Findings

Provides a pathwise solution notion for KPZ

Includes the Cole-Hopf solution as a special case

Establishes an approximation theory based on distribution regularization

Abstract

This work introduces a new notion of solution for the KPZ equation, in particular, our approach encompasses the Cole-Hopf solution. We set in the context of the distribution theory the proposed results by Bertini and Giacomin from the mid 90's. This new approach provides a pathwise notion of solution as well as a structured approximation theory. The developments are based on regularization arguments from the theory of distributions.