  A linear PDE problem for randomly perturbed domains is considered in an
adaptive Galerkin framework. The perturbation of the domain's boundary is
described by a vector valued random field depending on a countable number of
random variables in an affine way. The corresponding Karhunen-Lo\`eve expansion
is approximated by the pivoted Cholesky decomposition based on a prescribed
covariance function. The examined high-dimensional Galerkin system follows from
the domain mapping approach, transferring the randomness from the domain to the
diffusion coefficient and the forcing. In order to make this computationally
feasible, the representation makes use of the modern tensor train format for
the implicit compression of the problem. Moreover, an a posteriori error
estimator is presented, which allows for the problem-dependent iterative
refinement of all discretization parameters and the assessment of the achieved
error reduction. The proposed approach is demonstrated in numerical benchmark
problems.
