Inverse random source problem for Maxwell equations in an inhomogeneous medium

TL;DR

This work analyzes the inverse random source problem for time-harmonic Maxwell equations in an inhomogeneous medium driven by additive white noise. It establishes well-posedness and regularity for the direct problem using resolvent techniques in non-trapping media, and constructs a boundary-integral framework that connects the random source strength $\sigma$ to the tangential boundary data. Employing complex geometric optics (CGO) solutions and Ito isometry, the authors derive a logarithmic stability estimate for recovering $\sigma$ from single-frequency boundary measurements, with the rate depending on the Sobolev regularity $s$ of $\sigma$. The methodology is robust to inhomogeneity and extendable to other stochastic wave equations, offering a pathway for further study of stochastic inverse problems with rough sources and potential multi-component strengths.

Abstract

This paper concerns the inverse random source problem of the stochastic Maxwell equations driven by white noise in an inhomogeneous background medium. The well-posedness is established for the direct source problem, and the estimates and regularity of the solution are obtained. A logarithmic stability estimate is established for the inverse problem of determining the strength of the random source. The analysis only requires the random Dirichlet data at a fixed frequency.

Theorems & Definitions (11)

• Proposition 2.1
• Proposition 2.3
• Theorem 2.4
• Theorem 2.5
• proof
• Lemma 3.1
• proof
• Lemma 3.2
• Theorem 3.3
• proof