An invariance principle for stochastic heat equations with periodic coefficients

Abstract

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 to infinite-dimensional settings. Due to our results, $\frac{1}{\sqrt{t}}u(t,\cdot )$ converges weakly to a centered Gaussian variable whose covariance operator is described through Poisson 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 $\epsilon \downarrow 0$.

References in corpus (2)

• Proofs of the martingale FCLT
• The dynamical sine-Gordon model

Cited by in corpus (1)

• A central limit theorem for stochastic heat equations in random environment