  Generalized polynomial chaos expansions are a powerful tool to study
differential equations with random coefficients, allowing in particular to
efficiently approximate random invariant sets associated to such equations. In
this work, we use ideas from validated numerics in order to obtain rigorous a
posteriori error estimates together with existence results about gPC expansions
of random invariant sets. This approach also provides a new framework for
conducting validated continuation, i.e. for rigorously computing isolated
branches of solutions in parameter-dependent systems, which generalizes in a
straightforward way to multi-parameter continuation. We illustrate the proposed
methodology by rigorously computing random invariant periodic orbits in the
Lorenz system, as well as branches and 2-dimensional manifolds of steady states
of the Swift-Hohenberg equation.
