  We present an analysis of multilevel Monte Carlo techniques for the forward
problem of uncertainty quantification for the radiative transport equation,
when the coefficients ({\em cross-sections}) are heterogenous random fields. To
do this, we first give a new error analysis for the combined spatial and
angular discretisation in the deterministic case, with error estimates which
are explicit in the coefficients (and allow for very low regularity and jumps).
This detailed error analysis is done for the 1D space - 1D angle slab geometry
case with classical diamond differencing. Under reasonable assumptions on the
statistics of the coefficients, we then prove an error estimate for the random
problem in a suitable Bochner space. Because the problem is not self-adjoint,
stability can only be proved under a path-dependent mesh resolution condition.
This means that, while the Bochner space error estimate is of order
$\mathcal{O}(h^\eta)$ for some $\eta$, where $h$ is a (deterministically
chosen) mesh diameter, smaller mesh sizes might be needed for some
realisations. Under reasonable assumptions we show that the expected cost for
computing a typical quantity of interest remains of the same order as for a
single sample. This leads to rigorous complexity estimates for Monte Carlo and
multilevel Monte Carlo: For particular linear solvers, the multilevel version
gives up to two orders of magnitude improvement over Monte Carlo. We provide
numerical results supporting the theory.
