The KPZ Equation on the Real Line

TL;DR

This paper establishes the existence and uniqueness of solutions to the KPZ equation on the real line using paracontrolled analysis, and connects these solutions to directed polymer measures with a variational approach.

Contribution

It extends the analysis of the KPZ equation from the torus to the real line and introduces a path-by-path construction of the directed polymer measure.

Findings

Proves global existence and uniqueness of KPZ solutions on the real line.

Provides a comparison principle with quantitative bounds.

Constructs the directed polymer measure path-by-path and offers a variational characterization.

Abstract

We prove existence and uniqueness of distributional solutions to the KPZ equation globally in space and time, with techniques from paracontrolled analysis. Our main tool for extending the analysis on the torus to the full space is a comparison result that gives quantitative upper and lower bounds for the solution. We then extend our analysis to provide a path-by-path construction of the random directed polymer measure on the real line and we derive a variational characterisation of the solution to the KPZ equation.