Conditional GMC within the stochastic heat flow

TL;DR

This paper reveals that polymer measures associated with the stochastic heat flow exhibit a conditional Gaussian Multiplicative Chaos structure, linking different parameters and demonstrating positivity and convergence properties.

Contribution

It introduces a novel conditional GMC framework for the stochastic heat flow polymer measures, connecting measures with different parameters and establishing new properties.

Findings

Polymer measures have a conditional GMC structure.

The measure with noise strength a is law-equivalent to a shifted measure.

Polymer measures are almost surely strictly positive.

Abstract

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