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PreprintCC BY-NC-ND 4.0arXiv

Stability for the inverse source problem of the stochastic fractional Helmholtz equation

  • Dong Qiu
  • Xiang Xu
  • Yue Zhao
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Abstract

This paper is concerned with the inverse source problem for the stochastic fractional Helmholtz equation driven by white noise. For every fractional order 0<α<1, we prove the existence and uniqueness of the outgoing distributional solution to the direct problem with resolvent estimates at high frequencies, and establish its stochastic representation. For the inverse problem, we demonstrate that the variance of the source can be uniquely determined by the correlated random exterior data at a single frequency. We further establish an increasing stability estimate for the inverse problem by using the multifrequency correlated exterior data. Our stability result shows that as the upper bound of the bandwidth of the utilized frequency increases, the stability will also improve. The analysis employs the construction of geometric optics solutions, which connects the correlated data to the X-ray transform of the variance.

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Cite

BibTeXRIS
BibTeX
@misc{qiu2026stability,
  title = {{Stability for the inverse source problem of the stochastic fractional Helmholtz equation}},
  author = {Dong Qiu and Xiang Xu and Yue Zhao},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.27383},
  eprint = {2609.27383},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.27383},
  note = {CC BY-NC-ND 4.0},
}
RIS
TY  - UNPB
TI  - Stability for the inverse source problem of the stochastic fractional Helmholtz equation
AU  - Dong Qiu
AU  - Xiang Xu
AU  - Yue Zhao
PY  - 2026
DA  - 2026/09/23
PB  - arXiv
DO  - 10.48550/arxiv.2609.27383
UR  - https://arxiv.org/abs/2609.27383
AB  - This paper is concerned with the inverse source problem for the stochastic fractional Helmholtz equation driven by white noise. For every fractional order 0<α<1, we prove the existence and uniqueness of the outgoing distributional solution to the direct problem with resolvent estimates at high frequencies, and establish its stochastic representation. For the inverse problem, we demonstrate that the variance of the source can be uniquely determined by the correlated random exterior data at a single frequency. We further establish an increasing stability estimate for the inverse problem by using the multifrequency correlated exterior data. Our stability result shows that as the upper bound of the bandwidth of the utilized frequency increases, the stability will also improve. The analysis employs the construction of geometric optics solutions, which connects the correlated data to the X-ray transform of the variance.
ER  -