KPZ fluctuations in the planar stochastic heat equation
Probability
2025-09-09 v3 Mathematical Physics
Analysis of PDEs
math.MP
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.
Keywords
Cite
@article{arxiv.2210.13607,
title = {KPZ fluctuations in the planar stochastic heat equation},
author = {Jeremy Quastel and Alejandro Ramirez and Balint Virag},
journal= {arXiv preprint arXiv:2210.13607},
year = {2025}
}