{'fraction_non_alphanumeric': 0.052486187845303865, 'fraction_numerical': 0.008287292817679558, 'mean_word_length': 5.196581196581197, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
An invariance principle for stochastic heat equations with periodic coefficients

ABSTRACT 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, 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.