Exact Multi-Point Correlations in the Stochastic Heat Equation for Strictly Sublinear Coordinates

Abstract

We consider the Stochastic Heat Equation (SHE) in $(1+1)$ dimensions with delta Dirac initial data and spacetime white noise. We prove exact large-time asymptotics for multi-point correlations of the SHE for strictly sublinear space coordinates. The sublinear condition is optimal, in the sense that different asymptotics are known to occur when the space coordinates grow linearly [Lin 2023, Theorem 1.1]. Lastly, a notable feature of our result is that it confirms the connection between multi-point correlations in the SHE and the ground state of the Hamiltonian of the delta-Bose gas.