An inverse random source problem for the fractional Helmholtz equation

Peijun Li; Zhenqian Li

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arXiv · 2602.19559

An inverse random source problem for the fractional Helmholtz equation

Peijun Li·Zhenqian Li

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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.

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Peijun Li, Zhenqian Li. 2026-02-23. An inverse random source problem for the fractional Helmholtz equation. https://arxiv.org/abs/2602.19559

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