An inverse random source problem for the fractional Helmholtz equation
Abstract
This paper investigates an inverse random source problem for the stochastic fractional Helmholtz equation. The source is modeled as a centered, complex-valued, microlocally isotropic generalized Gaussian random field whose covariance and relation operators are described by classical pseudo-differential operators. For sufficiently large wavenumbers, we first establish the well-posedness of the direct problem in the distributional sense by analyzing the corresponding Lippmann--Schwinger integral equation. For the inverse problem, we show that the principal symbols of both the covariance and relation operators can be uniquely determined, with probability one, from the far-field patterns generated by a single realization of the random source. The approach employs a combination of the Born linearization, asymptotic expansions of the fractional Helmholtz Green kernel at high wavenumbers, and microlocal analysis of associated Fourier integral operators.
Cite
@article{arxiv.2602.19559,
title = {An inverse random source problem for the fractional Helmholtz equation},
author = {Peijun Li and Zhenqian Li},
journal= {arXiv preprint arXiv:2602.19559},
year = {2026}
}