Semimartingale characteristics of the two-dimensional stochastic heat equation at criticality

Abstract

We characterize the sample-path structure of the two-dimensional stochastic heat equation at criticality. The main theorems provide a complete, pathwise description of its semimartingale characteristics and establish explicit, exact formulas for the covariation measure. The proof derives an asymptotic expansion for the covariation measures of regularized approximate solutions by integrating techniques from quantum-solvable operators into stochastic integration theory.