An invariance principle for stochastic heat equations with periodic coefficients

Lu Xu

Published 2015-05-13Version 1

In this paper we investigate the asymptotic behaviors of solution $u(t, \cdot)$ of a stochastic heat equation with a periodic nonlinear term. Such equation appears related to the dynamical sine-Gordon model. We consider the reversible case and extend the central limit theorem for diffusions in finite dimensions 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 explicitly described. Different from the finite dimensional case, the fluctuation in $x$ 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 $\epsilon \downarrow 0$.