  Determining the flow of rays or particles driven by a force or velocity field
is fundamental to modelling many physical processes, including weather
forecasting and the simulation of molecular dynamics. High frequency wave
energy distributions can also be approximated using flow or transport
equations. Applications arise in underwater and room acoustics,
vibro-acoustics, seismology, electromagnetics, quantum mechanics and in
producing computer generated imagery. In many practical applications, the
driving field is not known exactly and the dynamics are determined only up to a
degree of uncertainty. This paper presents a boundary integral framework for
propagating flows including uncertainties, which is shown to systematically
interpolate between a deterministic and a completely random description of the
trajectory propagation. A simple but efficient discretisation approach is
applied to model uncertain billiard dynamics in an integrable rectangular
domain.
