Inverse source problem for the stochastic fractional Helmholtz equation in inhomogeneous media
Abstract
This paper is concerned with an inverse source problem for the stochastic fractional Helmholtz equation in a three-dimensional inhomogeneous medium. The source is modeled as real Gaussian spatial white noise with an unknown nonnegative strength function. For admissible frequencies, we establish well-posedness of the radiating solution in the distributional sense and show that it has a smooth representation outside the source region. For the inverse problem, by using two-point correlations of the field measured in an exterior open set at a single frequency, we prove a logarithmic stability estimate for recovering the source strength under an a priori Sobolev bound. As a uniform Hölder stability estimate is unavailable for the general admissible class, by assuming the difference between two source strengths has a fixed sign, we improve the stability to Hölder continuity with an optimal Hölder exponent. The analysis combines scattering theory and quantitative continuation for the fractional Laplacian using the Caffarelli-Silvestre extension.
Related
- Stability for the inverse source problem of the stochastic fractional Helmholtz equation
- Increasing stability for the inverse potential problem of the fractional Helmholtz equation
- An Inverse Random Source Problem for the Moore-Gibson-Thompson Equation Driven by Fractional Brownian Motion
- Inverse Source Problem for a Time-Fractional Diffusion-Wave Equation with a Singular Inverse-Square Potential
- Space-time fractional stochastic Burgers-type equation
BibTeX
@misc{qiu2026inverse,
title = {{Inverse source problem for the stochastic fractional Helmholtz equation in inhomogeneous media}},
author = {Dong Qiu and Xiang Xu and Yue Zhao},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.41928},
eprint = {2610.41928},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.41928},
}
RIS
TY - UNPB
TI - Inverse source problem for the stochastic fractional Helmholtz equation in inhomogeneous media
AU - Dong Qiu
AU - Xiang Xu
AU - Yue Zhao
PY - 2026
DA - 2026/10/30
PB - arXiv
DO - 10.48550/arxiv.2610.41928
UR - https://arxiv.org/abs/2610.41928
AB - This paper is concerned with an inverse source problem for the stochastic fractional Helmholtz equation in a three-dimensional inhomogeneous medium. The source is modeled as real Gaussian spatial white noise with an unknown nonnegative strength function. For admissible frequencies, we establish well-posedness of the radiating solution in the distributional sense and show that it has a smooth representation outside the source region. For the inverse problem, by using two-point correlations of the field measured in an exterior open set at a single frequency, we prove a logarithmic stability estimate for recovering the source strength under an a priori Sobolev bound. As a uniform Hölder stability estimate is unavailable for the general admissible class, by assuming the difference between two source strengths has a fixed sign, we improve the stability to Hölder continuity with an optimal Hölder exponent. The analysis combines scattering theory and quantitative continuation for the fractional Laplacian using the Caffarelli-Silvestre extension.
ER -