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PreprintCC BY 4.0arXiv

Chaotic CLT using Poisson Fourth Moment Theorem and Graph Theory

  • Raluca M. Balan
  • Jaime Garza
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Abstract

In this article, we consider the parabolic Anderson model and the hyperbolic Anderson model in dimension 1, driven by a time-independent Lévy noise, which is either white, or colored with integrable coloration kernel κ. Using a Poisson version of the Fourth Moment Theorem due to Döbler, Vidotto and Zheng (2018), we prove that the spatial integral of the solution satisfies a central limit theorem. We prove that this result also holds for the hyperbolic Anderson model in dimension d=2, if in addition κ is bounded. To estimate the fourth cumulant of the projection of the spatial integral on a fixed chaos, we develop a novel technique using graph theory, which allows us to handle some intricate integrals involving the modified contractions.

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Cite

BibTeXRIS
BibTeX
@misc{balan2026chaotic,
  title = {{Chaotic CLT using Poisson Fourth Moment Theorem and Graph Theory}},
  author = {Raluca M. Balan and Jaime Garza},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.27299},
  eprint = {2610.27299},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.27299},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Chaotic CLT using Poisson Fourth Moment Theorem and Graph Theory
AU  - Raluca M. Balan
AU  - Jaime Garza
PY  - 2026
DA  - 2026/10/19
PB  - arXiv
DO  - 10.48550/arxiv.2610.27299
UR  - https://arxiv.org/abs/2610.27299
AB  - In this article, we consider the parabolic Anderson model and the hyperbolic Anderson model in dimension 1, driven by a time-independent Lévy noise, which is either white, or colored with integrable coloration kernel κ. Using a Poisson version of the Fourth Moment Theorem due to Döbler, Vidotto and Zheng (2018), we prove that the spatial integral of the solution satisfies a central limit theorem. We prove that this result also holds for the hyperbolic Anderson model in dimension d=2, if in addition κ is bounded. To estimate the fourth cumulant of the projection of the spatial integral on a fixed chaos, we develop a novel technique using graph theory, which allows us to handle some intricate integrals involving the modified contractions.
ER  -