KPZ fluctuations in the planar stochastic heat equation

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 Feynman–Kac formula and defines a randomized shift, or Gaussian multiplicative chaos (GMC). The fluctuations far from the center are shown to be given by the one-dimensional Kardar–Parisi–Zhang (KPZ) equation.