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We consider the stochastic heat equation on $\mathbb R^d$ with multiplicativespace-time white noise noise smoothed in space. For $d\geq 3$ and small noiseintensity, the solution is known to converge to a strictly positive randomvariable as the smoothing parameter vanishes. In this regime, we study the rateof convergence and show that the pointwise fluctuations of the smoothenedsolutions as well as that of the underlying martingale of the Brownian directedpolymer converge to a Gaussian limit.

Fluctuation and Rate of Convergence for the Stochastic Heat Equation in Weak Disorder