Conditional GMC within the stochastic heat flow

We establish that the family of polymer measures $M^θ_{[s,t]}$ associated with the Stochastic Heat Flow (SHF), indexed by $θ\in\mathbb{R}$, has a conditional Gaussian Multiplicative Chaos (GMC) structure. Namely, taking the random measure $M^θ_{[s,t]}$ 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 $M^{θ+a}_{[s,t]}$. 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 $θ\to\infty$.