24. chen.khoshnevisan.ea:22:spatial

Spatial ergodicity and central limit theorems for parabolic Anderson model with delta initial condition

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

Abstract: Let \(\{u(t\,, x)\}_{t >0, x \in\mathbb{R}}\) denote the solution to the parabolic Anderson model with initial condition \(\delta_0\) and driven by space-time white noise on \(\mathbb{R}_+\times\mathbb{R}\), and let \(p_t(x):= (2\pi t)^{-1/2}\exp\{-x^2/(2t)\}\) denote the standard Gaussian heat kernel on the line. We use a non-trivial adaptation of the methods in our companion papers [CKNP21b, CKNP22a] in order to prove that the random field \(x\mapsto u(t\,,x)/p_t(x)\) is ergodic for every \(t >0\). And we establish an associated quantitative central limit theorem following the approach based on the Malliavin-Stein method introduced in Huang, Nualart, and Viitasaari [HNV20].

MSC 2010 subject classifications: Primary. 60H15; Secondary. 60H07, 60F05.

Keywords: parabolic Anderson model, ergodicity, central limit theorem, Malliavin calculus, Stein’s method, Dirac delta initial condition.

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[CKNP22b] Chen, Le, Khoshnevisan, Davar, Nualart, David & Pu, Fei (2022) ‘Spatial ergodicity and central limit theorems for parabolic Anderson model with delta initial condition’, J. Funct. Anal. 282, Paper No. 109290, 35. <https://doi.org/10.1016/j.jfa.2021.109290>

@article{chen.khoshnevisan.ea:22:spatial,
  author        = {Chen, Le and Khoshnevisan, Davar and Nualart, David and Pu, Fei},
  title         = {Spatial ergodicity and central limit theorems for parabolic {A}nderson model with delta initial condition},
  journal       = {J. Funct. Anal.},
  fjournal      = {Journal of Functional Analysis},
  volume        = {282},
  year          = {2022},
  number        = {2},
  pages         = {Paper No. 109290, 35},
  issn          = {0022-1236},
  mrclass       = {60H15 (60F05 60H07)},
  mrnumber      = {4334682},
  mrreviewer    = {Ciprian A. Tudor},
  doi           = {10.1016/j.jfa.2021.109290},
  url           = {https://doi.org/10.1016/j.jfa.2021.109290}
}

References: Amir et al. [ACQ11]; Balan et al. [BNQSZ22]; Billingsley [Bil99]; Chen and Dalang [CD15b]; Chen et al. [CHN21]; Chen and Huang [CH19a]; Chen et al. [CKNP21b]; Chen et al. [CKNP22a]; Conus et al. [CJKS13b]; Delgado-Vences et al. [DVNZ20]; Federer [Fed69]; Huang et al. [HNV20]; Huang et al. [HNVZ20]; Nourdin and Peccati [NP09]; Nourdin and Peccati [NP12]; Nualart [Nua09]; Nualart and Nualart [NN18]; Walsh [Wal86];

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