Stochastic heat flow by moments

Abstract: The critical two-dimensional Stochastic Heat Flow (SHF) is the scaling limit of the directed polymers in random environments and the noise-mollified Stochastic Heat Equation (SHE), at the critical dimension of two and near the critical temperature. The work of Caravenna, Sun, and Zygouras (2023) proved that the discrete polymers converge to a universal (model-independent) limit, thereby identifying the limit as the SHF. In this work, we take a different approach: We formulate a set of axioms for the SHF, prove the uniqueness in law under these axioms, and also prove the existence by showing that the noise-mollified SHE converges to the SHF under this formulation. The set consists of three axioms commonly seen in stochastic flows and one axiom about the first four moments of the SHF over every time interval.