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


We build a regular version of a five parameter field which describes the Green’s function, or fundamental solution, of the parabolic Anderson model (PAM) with white noise forcing on (1+1)-dimensional space-time. Through the superposition principle, our construction gives a pointwise coupling of all solutions to the PAM with initial or terminal conditions satisfying sharp growth assumptions, for all initial and terminal times. Using this coupling, we show that the PAM with a (sub-)exponentially growing initial condition admits conserved quantities given by the limiting exponential slope, in addition to proving many new basic properties of solutions to the PAM with general initial conditions. These properties are then connected to the existence, regularity, and continuity of the quenched continuum polymer measures. Through the polymer connection, we also show that the kernel is strictly totally positive.

Tom Alberts
Tom Alberts
Associate Professor of Mathematics
University of Utah