The Green’s function of the parabolic Anderson model and the continuum directed polymer

We study the behavior of the coupling of solutions to the stochastic heat equation/parabolic Anderson model (PAM) obtained through convolution with the Green’s function. In this coupling, we show that the PAM with a (sub-)exponentially growing initial condition has conserved quantities given by the slopes at ∞ ∞ and -∞ - ∞ . These are then connected to Hopf–Cole solutions to the KPZ equation and the existence, regularity, and continuity of the quenched continuum polymer measures. We also show that the Green’s function is strictly totally positive using a novel coupling argument based on the continuum directed polymer.