  Polynomial chaos expansions (PCE) allow us to propagate uncertainties in the
coefficients of differential equations to the statistics of their solutions.
Their main advantage is that they replace stochastic equations by systems of
deterministic equations. Their main challenge is that the computational cost
becomes prohibitive when the dimension of the parameters modeling the
stochasticity is even moderately large. We propose a generalization of the PCE
framework that allows us to keep this dimension as small as possible in
favorable situations. For instance, in the setting of stochastic differential
equations (SDEs) with Markov random forcing, we expect the future evolution to
depend on the present solution and the future stochastic variables. We present
a restart procedure that precisely allows PCE to depend only on that
information. The computational difficulty then becomes the construction of
orthogonal polynomials for dynamically evolving measures. We present
theoretical results of convergence for our Dynamical generalized Polynomial
Chaos (DgPC) method. Numerical simulations for linear and nonlinear SDEs show
that it adequately captures the long-time behavior of their solutions as well
as their invariant measures when the latter exist.
