Delta-Bose gas from the viewpoint of the two-dimensional stochastic heat equation

Abstract

In order to give a dual, annealed description of the two-dimensional stochastic heat equation (SHE) from regularizing the noise, we consider the Schrödinger semigroup of the many-body delta-Bose gas from mollifying the delta potentials. The main theorem proves the convergences of the corresponding approximate semigroups when they act on bounded functions. For the proof we introduce a mean-field Poisson system to expand the Feynman–Kac formula of the approximate semigroups. This expansion yields infinite series, showing certain Markovian decompositions of the summands into nonconcurrent, nonconsecutive two-body interactions. Components in these decompositions are then grouped nonlinearly in time to establish the dominated convergence of the infinite series. With regards to the two-dimensional SHE, the main theorem also characterizes the Nth moments for all $\mathit{N}\ge 3$ under any bounded initial condition. A particular example is the constant initial condition under which the solution of the SHE has the interpretation of the partition function of the continuum directed random polymer.