  This work considers the problem of numerically approximating statistical
moments of a Quantity of Interest (QoI) that depends on the solution of a
linear parabolic partial differential equation. The geometry is assumed to be
random and is parameterized by $N$ random variables. The parabolic problem is
remapped to a fixed deterministic domain with random coefficients and shown to
admit an extension on a well defined region embedded in the complex hyperplane.
A Stochastic collocation method with an isotropic Smolyak sparse grid is used
to compute the statistical moments of the QoI. In addition, convergence rates
for the stochastic moments are derived and compared to numerical experiments.

Numerical Analysis