KPZ fluctuations in the planar stochastic heat equation

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.

References in corpus (3)

• Probability Distribution of the Free Energy of the Continuum Directed Random Polymer in 1+1 dimensions
• Self-intersection local time: Critical exponent, large deviations, and laws of the iterated logarithm
• Exponential asymptotics and law of the iterated logarithm for intersection local times of random walks