  In this work, we consider the propagation of acoustic waves in unbounded
domains characterized by a constant wavenumber, except possibly in a bounded
region. The geometry of this inhomogeneity is assumed to be uncertain, and we
are particularly interested in studying the propagation of this behavior
throughout the physical model considered. A key step in our analysis consists
of recasting the physical model-originally set in an unbounded domain-into a
computationally manageable formulation based on Volume Integral Equations
(VIEs), particularly the Lippmann-Schwinger equation. We show that both the
leading operator in this volume integral formulation and its solution depend
holomorphically on shape variations of the support of the aforementioned
inhomogeneity. This property, known as shape holomorphy, is crucial in the
analysis and implementation of various methods used in computational
Uncertainty Quantification (UQ). We explore the implications of this result in
forward and inverse UQ and provide numerical experiments illustrating and
confirming the theoretical predictions.
