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A Central Limit Theorem for Stochastic Heat Equations in Random Environment

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Abstract

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.

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Acknowledgements

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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  1. Graduate School of Mathematical Sciences, The University of Tokyo, Tokyo, Japan

    Lu Xu

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  1. Lu Xu
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Correspondence to Lu Xu.

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Xu, L. A Central Limit Theorem for Stochastic Heat Equations in Random Environment. J Theor Probab 31, 1356–1379 (2018). https://doi.org/10.1007/s10959-017-0748-2

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  • DOI: https://doi.org/10.1007/s10959-017-0748-2

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