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Stochastic PDE construction of some non-local or non-translation-invariant Anderson operators in the full subcritical regime

  • Léonard Ferdinand
  • Samaël Mackowiak
  • Aurélien Minguella
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Abstract

We construct Anderson-type operators in the full subcritical regime via a SPDE argument. Our work encompasses perturbations by a Gaussian noise of the fractional Laplacian on the torus or on the full space, and of the harmonic oscillator on the full space. In every case, we are able to derive an optimal estimate on the difference between the solution and the deterministic heat flow, which allows us to get a Weyl law for the fractional case on the torus and for the harmonic oscillator case, and so in the full subcritical regime. For the non-translation-invariant operator, the main novelty is that we develop a general argument in order to prove how to renormalise the operator with some constants. The SPDE argument uses the flow approach introduced by Duch. To construct the operators, we use the Hille-Yosida theorem for the fractional Laplacian on the torus and the harmonic oscillator, and the Klein-Landau theorem for the fractional Laplacian in the full space.

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Cite

BibTeXRIS
BibTeX
@misc{ferdinand2026stochastic,
  title = {{Stochastic PDE construction of some non-local or non-translation-invariant Anderson operators in the full subcritical regime}},
  author = {Léonard Ferdinand and Samaël Mackowiak and Aurélien Minguella},
  howpublished = {HAL},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.05391},
  eprint = {2610.05391},
  archivePrefix = {arXiv},
  url = {https://hal.science/hal-05781089v1},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Stochastic PDE construction of some non-local or non-translation-invariant Anderson operators in the full subcritical regime
AU  - Léonard Ferdinand
AU  - Samaël Mackowiak
AU  - Aurélien Minguella
PY  - 2026
DA  - 2026/10/04
PB  - HAL
DO  - 10.48550/arxiv.2610.05391
UR  - https://hal.science/hal-05781089v1
AB  - We construct Anderson-type operators in the full subcritical regime via a SPDE argument. Our work encompasses perturbations by a Gaussian noise of the fractional Laplacian on the torus or on the full space, and of the harmonic oscillator on the full space. In every case, we are able to derive an optimal estimate on the difference between the solution and the deterministic heat flow, which allows us to get a Weyl law for the fractional case on the torus and for the harmonic oscillator case, and so in the full subcritical regime. For the non-translation-invariant operator, the main novelty is that we develop a general argument in order to prove how to renormalise the operator with some constants. The SPDE argument uses the flow approach introduced by Duch. To construct the operators, we use the Hille-Yosida theorem for the fractional Laplacian on the torus and the harmonic oscillator, and the Klein-Landau theorem for the fractional Laplacian in the full space.
ER  -