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Central limit theorems for stochastic heat equation driven by Gaussian noise with heat-kernel covariance

  • Meng Wang
  • Wangjun Yuan
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Abstract

In this article, we study the spatial fluctuations of the Skorohod solution to a stochastic heat equation on ℝd for d<4, driven by multiplicative Gaussian noise with the non-separable covariance kernel p|t−s|(x−y). We prove that the solution is strictly stationary and spatially ergodic at every fixed time. For the centered spatial integral FR(t)=∫{|x|<R}(u(t,x)−1)dx, we show that 𝔼[FR(t)FR(s)]∼ K(t,s)Rd as R→∞. Using moment estimates for the first two Malliavin derivatives and a second-order Gaussian Poincaré inequality, we establish a quantitative central limit theorem in total variation distance with rate R−d/2. We also prove a functional central limit theorem for the process {R−d/2FR(t)}t≥0.

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Cite

BibTeXRIS
BibTeX
@misc{wang2026central,
  title = {{Central limit theorems for stochastic heat equation driven by Gaussian noise with heat-kernel covariance}},
  author = {Meng Wang and Wangjun Yuan},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.04941},
  eprint = {2610.04941},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.04941},
}
RIS
TY  - UNPB
TI  - Central limit theorems for stochastic heat equation driven by Gaussian noise with heat-kernel covariance
AU  - Meng Wang
AU  - Wangjun Yuan
PY  - 2026
DA  - 2026/10/04
PB  - arXiv
DO  - 10.48550/arxiv.2610.04941
UR  - https://arxiv.org/abs/2610.04941
AB  - In this article, we study the spatial fluctuations of the Skorohod solution to a stochastic heat equation on ℝ^d for d<4, driven by multiplicative Gaussian noise with the non-separable covariance kernel p_|t−s|(x−y). We prove that the solution is strictly stationary and spatially ergodic at every fixed time. For the centered spatial integral F_R(t)=∫_{|x|<R}(u(t,x)−1)dx, we show that 𝔼[F_R(t)F_R(s)]∼ K(t,s)R^d as R→∞. Using moment estimates for the first two Malliavin derivatives and a second-order Gaussian Poincaré inequality, we establish a quantitative central limit theorem in total variation distance with rate R^−d/2. We also prove a functional central limit theorem for the process {R^−d/2F_R(t)}_t≥0.
ER  -