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.
KeywordsStochastic 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.
- 2.Bogachev, V.I.: Gaussian Measures, Mathematical Surveys and Monographs, vol. 62. American Mathematical Society, Providence (2011)Google Scholar
- 13.Osada, H.: Homogenization of diffusion processes with random stationary coefficients. In: Probability Theory and Mathematical Statistics, Tbilissi, 1982. Lect. Notes Math, vol. 1021, pp. 507–517 (1983)Google Scholar
- 16.Papanicolaou, G., Varadhan, S.R.S.: Boundary value problems with rapidly oscillating random coefficients, random fields, vols I, II, Esztergom, 1979. Colloq Math. Soc. János Bolyai 27, 835–873 (1981)Google Scholar
- 17.Papanicolaou, G., Varadhan, S.R.S.: Diffusions with random coefficients. In: Kallianpur G., Krishnaiah P.R., Ghosh J.K. (eds.) Statistics and Probability: Essays in Honor of C.R. Rao. North-Holland Pub. Co. pp. 547–552 (1982)Google Scholar
- 20.Xu, L.: An invariance principle for stochastic heat equations with periodic coefficients, available at arXiv:1505.03391 (2015)
- 21.Yosida, K.: Functional Analysis, Classics in Mathematics. Springer, Berlin (1980)Google Scholar