21. chen.khoshnevisan.ea:22:central

Central limit theorems for parabolic stochastic partial differential equations

Le Chen, Davar Khoshnevisan, David Nualart, and Fei Pu

Abstract: Let \(\{u(t\,,x)\}_{t\ge 0, x\in \mathbb{R}^d}\) denote the solution of a \(d\)-dimensional nonlinear stochastic heat equation that is driven by a Gaussian noise, white in time with a homogeneous spatial covariance that is a finite Borel measure \(f\) and satisfies Dalang’s condition. We prove two general functional central limit theorems for occupation fields of the form

\[N^{-d} \int_{\mathbb{R}^d} g(u(t\,,x)) \psi(x/N)\,d x \quad \text{as}\quad N\rightarrow \infty,\]

where \(g\) runs over the class of Lipschitz functions on \(\mathbb{R}^d\) and \(\psi\in L^2(\mathbb{R}^d)\). The proof uses Poincaré-type inequalities, Malliavin calculus, compactness arguments, and Paul Lévy’s classical characterization of Brownian motion as the only mean zero, continuous Lévy process. Our result generalizes central limit theorems of Huang et al [HNV20, HNVZ20] valid when \(g(u)=u\) and \(\psi = \mathbf{1}_{[0,1]^d}\).

MSC 2010 subject classifications: Primary 60H15; Secondary 60F17, 60H07.

Keywords: stochastic heat equation, central limit theorem, Poincaré inequalities, Malliavin calculus, metric entropy.

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[CKNP22a] Chen, Le, Khoshnevisan, Davar, Nualart, David & Pu, Fei (2022) ‘Central limit theorems for parabolic stochastic partial differential equations’, Ann. Inst. Henri Poincar’e Probab. Stat. 58, 1052–1077. <https://doi.org/10.1214/21-aihp1189>

@article{chen.khoshnevisan.ea:22:central,
  author        = {Chen, Le and Khoshnevisan, Davar and Nualart, David and Pu, Fei},
  title         = {Central limit theorems for parabolic stochastic partial differential equations},
  journal       = {Ann. Inst. Henri Poincar\'{e} Probab. Stat.},
  fjournal      = {Annales de l'Institut Henri Poincar\'{e} Probabilit\'{e}s et Statistiques},
  volume        = {58},
  year          = {2022},
  number        = {2},
  pages         = {1052--1077},
  issn          = {0246-0203},
  mrclass       = {60H15 (60F17 60H07)},
  mrnumber      = {4421618},
  doi           = {10.1214/21-aihp1189},
  url           = {https://doi.org/10.1214/21-aihp1189}
}

References: Bertoin [Ber96]; Bradley [Bra07]; Burkholder et al. [BDG72]; Carlen and Krée [CK91]; Chen and Huang [CH19a]; Chen et al. [CKNP21b]; Dalang [Dal99]; Davis [Dav76]; Doob [Doo90]; Dym and McKean [DM76]; Federer [Fed69]; Foondun and Khoshnevisan [FK13]; Huang et al. [HNV20]; Huang et al. [HNVZ20]; Karatzas and Shreve [KS91]; Khoshnevisan [Kho14]; Marcus and Rosen [MR06]; Maruyama [Mar49]; Nualart [Nua09]; Volkonskiui and Rozanov [VR59]; Walsh [Wal86];

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