Conditional GMC within the stochastic heat flow

Abstract

We establish that the family of polymer measures $M^{\hat{I}}{¸}_{[s,t]}$ associated with the Stochastic Heat Flow (SHF), indexed by $\hat{I}¸\in \mathbb{R}$, has a conditional Gaussian Multiplicative Chaos (GMC) structure. Namely, taking the random measure $M^{\hat{I}}{¸}_{[s,t]}$ as the reference measure, we construct the path-space GMC with noise strength $a>0$ and prove that the resulting random measure is equal in law to $M_{[s,t]}^{\hat{I}¸+a}$. As two applications, we prove that the polymer measure and SHF tested against general nonnegative functions are almost surely strictly positive and that the SHF converges to $0$ as $\hat{I}¸\to \infty$.

42 pages