Stability for inverse random source problems of the polyharmonic wave equation

TL;DR

This paper establishes stability estimates for inverse source problems involving stochastic polyharmonic wave equations with white noise sources, providing both Hölder and logarithmic stability results depending on the presence of a potential.

Contribution

It introduces new stability estimates for inverse stochastic polyharmonic wave problems, including Hölder and logarithmic types, based on source regularity and potential conditions.

Findings

Hölder stability without potential

Logarithmic stability with potential

Well-posedness of the direct problem

Abstract

This paper investigates stability estimates for inverse source problems in the stochastic polyharmonic wave equation, where the source is represented by white noise. The study examines the well-posedness of the direct problem and derives stability estimates for identifying the strength of the random source. Assuming a priori information of the regularity and support of the source strength, the H\"{o}lder stability is established in the absence of a potential. In the more challenging case where a potential is present, the logarithmic stability estimate is obtained by constructing specialized solutions to the polyharmonic wave equation.