  The present work is devoted to approximation of the statistical moments of
the unknown solution of a class of elliptic transmission problems in $\mathbb
R^3$ with randomly perturbed interfaces. Within this model, the diffusion
coefficient has a jump discontinuity across the random transmission interface
which models linear diffusion in two different media separated by an uncertain
surface. We apply the shape calculus approach to approximate solution's
perturbation by the so-called shape derivative, correspondingly statistical
moments of the solution's perturbation are approximated by the moments of the
shape derivative. We characterize the shape derivative as a solution of a
related homogeneous transmission problem with nonzero jump conditions which can
be solved with the aid of boundary integral equations. We develop a rigorous
theoretical framework for this method, particularly i) extending the method to
the case of unbounded domains and ii) closing the gaps and clarifying and
adapting results in the existing literature. The theoretical findings are
supported by and illustrated in two particular examples.
