Time-dependent averages of a critical long-range stochastic heat equation

We study the time-dependent spatial averages of a critical stochastic partial differential equation, namely the stochastic heat equation in dimension $d\ge 3$ with noise white in time and colored in space with covariance kernel $\mathrm{\parallel }\cdot {\mathrm{\parallel }}^{-2}$. The solution to this SPDE is a singular measure and was constructed by Mueller and Tribe in [MT04]. We show that the time-dependent spatial averages of this SPDE over a ball of radius $R$ at time $t$ have different limits under different space-time scales. In particular, when $t\ll R^2$, the central limit theorem holds; when $t=R^2$, the spatial average is a non-Gaussian random variable; when $t\gg R^2$, the spatial average becomes extinct.