Work

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

Year: 2026

Type: article

Abstract: 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 $$\infty $$ ∞ and $$-\infty $$ - ∞ . 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.

Source: Stochastic Partial Differential Equations Analysis and Computations

DOI: 10.1007/s40072-026-00440-0

Authors: Tom Alberts, Christopher Janjigian, Firas Rassoul-Agha, Timo Seppäläinen

Language: English

Topic: Stochastic processes and statistical mechanics

Subfield: Mathematical Physics

Field: Mathematics

Domain: Physical Sciences

Open Access status: hybrid

Estimated APC paid: $3,190

Funders: National Science Foundation, Simons Foundation, Directorate for Mathematical and Physical Sciences