KPZ fluctuations in the planar stochastic heat equation

2022/10/24 by Jeremy Quastel, Quastel, Jeremy, Alejandro F. Ramı́rez +3 · 1 citation
Mathematics · Physics and Astronomy · #Advanced Thermodynamics and Statistical Mechanics #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR) #Stochastic processes and statistical mechanics #Theoretical and Computational Physics

paper · pdf · doi:10.48550/arxiv.2210.13607

openalex publication_date 2022/10/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We use a version of the Skorokhod integral to give a simple and rigorous formulation of the Wick-ordered (stochastic) heat equation with planar white noise, representing the free energy of an undirected random polymer. The solution for all times is expressed as the L1 limit of a martingale given by the Feyman-Kac formula and defines a randomized shift, or Gaussian multiplicative chaos. The fluctuations far from the centre are shown to be given by the one-dimensional KPZ equation.

Cited by

• Conditional GMC within the stochastic heat flow

Related

• The two-dimensional KPZ equation in the entire subcritical regime
• Fluctuation and Rate of Convergence for the Stochastic Heat Equation in Weak Disorder
• Fluctuations of two-dimensional stochastic heat equation and KPZ equation in subcritical regime for general initial conditions
• Free energy fluctuations for directed polymers in random media in 1+1\n dimension
• Hierarchy of KPZ limits arising from directed random walk models in random media