Gaussian fluctuations for the stochastic heat equation with colored noise

  • Jingyu Huang
  • David Nualart
  • Lauri Viitasaari
  • Guangqu ZhengEmail author


In this paper, we present a quantitative central limit theorem for the d-dimensional stochastic heat equation driven by a Gaussian multiplicative noise, which is white in time and has a spatial covariance given by the Riesz kernel. We show that the spatial average of the solution over an Euclidean ball is close to a Gaussian distribution, when the radius of the ball tends to infinity. Our central limit theorem is described in the total variation distance, using Malliavin calculus and Stein’s method. We also provide a functional central limit theorem.


Stochastic heat equation Central limit theorem Malliavin calculus Stein’s method 

Mathematics Subject Classification

60H15 60H07 60G15 60F05 



The authors thank the anonymous referee for many constructive advices that improved this paper.


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Authors and Affiliations

  1. 1.School of MathematicsUniversity of BirminghamBirminghamUK
  2. 2.Department of MathematicsUniversity of KansasLawrenceUSA
  3. 3.Department of Mathematics and StatisticsUniversity of HelsinkiHelsinkiFinland

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