Gaussian fluctuations for the stochastic heat equation with colored noise


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.

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

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Correspondence to Guangqu Zheng.

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David Nualart is supported by NSF Grant DMS 1811181.

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Huang, J., Nualart, D., Viitasaari, L. et al. Gaussian fluctuations for the stochastic heat equation with colored noise. Stoch PDE: Anal Comp 8, 402–421 (2020).

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

Mathematics Subject Classification

  • 60H15
  • 60H07
  • 60G15
  • 60F05