Stochastic heat flow by moments

TL;DR

This paper introduces a new formulation and characterization of the Stochastic Heat Flow (SHF), a process arising as a limit of directed polymers and the SHE, by matching moments up to the fourth order.

Contribution

It provides an independent approach to defining the SHF, establishing its existence and uniqueness through a set of axioms based on moment matching.

Findings

Established the existence of the SHF under the new axioms.

Proved the uniqueness of the SHF with the moment-matching axioms.

Provided a continuous process formulation of the SHF.

Abstract

The Stochastic Heat Flow (SHF) emerges as the scaling limit of directed polymers in random environments and the noise-mollified Stochastic Heat Equation (SHE), specifically at the critical dimension of two and near the critical temperature. The prior work Caravenna Sun Zygouras (2023) established the first construction of finite-dimensional distributions by demonstrating the universal (model-independent) convergence of discrete polymers. In this work, we present a new, independent approach to the SHF. We formulate the SHF as a continuous process and provide a set of axioms for its characterization. We establish both the uniqueness and existence of this process under our new formulation, with a key feature of these axioms being the matching of the first four moments.