Journal of Theoretical Probability

, Volume 31, Issue 3, pp 1356–1379 | Cite as

A Central Limit Theorem for Stochastic Heat Equations in Random Environment



In this article, we investigate the asymptotic behavior of the solution to a one-dimensional stochastic heat equation with random nonlinear term generated by a stationary, ergodic random field. We extend the well-known central limit theorem for finite-dimensional diffusions in random environment to this infinite-dimensional setting. Due to our result, a central limit theorem in \(L^1\) sense with respect to the randomness of the environment holds under a diffusive time scaling. The limit distribution is a centered Gaussian law whose covariance operator is explicitly described. The distribution concentrates only on the space of constant functions.


Stochastic heat equation Random environment Central limit theorem 

Mathematics Subject Classification (2010)

60F05 60H15 60K37 



The author greatly thanks Professor Tadahisa Funaki and Professor Stefano Olla for their instructive discussion and suggestions. The author also thanks Professor Jean-Dominique Deuschel for his comments on quenched results.


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© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Graduate School of Mathematical SciencesThe University of TokyoTokyoJapan

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