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, 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 Mθ+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→∞.