An invariance principle for stochastic heat equations with periodic coefficients
Lu Xu · Stochastic Analysis and Applications · 2017
We investigate the asymptotic behaviors of the solution u(t, ·) to a stochastic heat equation with a periodic, gradient-type nonlinear term. We extend the central limit theorem for finite-dimensional diffusions presented in [1 Komorowski, T., Landim, C., Olla, S. (2012). Fluctuations in Markov processes. Time symmetry and martingale approximation. In: A. Chenciner J. Coates and S.R.S. Varadhan eds. Grundlehren der Mathematischen Wissenschaften. Vol. 345, pp. xviii+494. Heidelberg: Springer-Verlag. [Google Scholar], sec. 9.1] to infinite-dimensional settings. Due to our results, as t → ∞, 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 {εu(ε− 2t, ·)}t ∈ [0, T] as ε↓0.