An invariance principle for stochastic heat equations with periodic coefficients

We investigate the asymptotic behaviors of the solution $u(t,\cdot)$ to a stochastic heat equation with a periodic, gradient-type nonlinear term. We extend the central limit theorem for finite-dimensional diffusions presented in \cite[\S9.1]{KLO12} to infinite-dimensional settings. Due to our results, as $t\rightarrow\infty$, $\frac{1}{\sqrt{t}}u(t,\cdot)$ converges weakly to a centered Gaussian variable whose covariance operator is described through Poisson's equations. Different from the finite-dimensional case, the fluctuation in space vanishes in the limit distribution. Furthermore, we verify the tightness and present an invariance principle for $\{\epsilon u(\epsilon^{-2}t,\cdot)\}_{t \in[0,T]}$ as $\epsilon \downarrow 0$.