  This letter presents a non-parametric modeling approach for forecasting
stochastic dynamical systems on low-dimensional manifolds. The key idea is to
represent the discrete shift maps on a smooth basis which can be obtained by
the diffusion maps algorithm. In the limit of large data, this approach
converges to a Galerkin projection of the semigroup solution to the underlying
dynamics on a basis adapted to the invariant measure. This approach allows one
to quantify uncertainties (in fact, evolve the probability distribution) for
non-trivial dynamical systems with equation-free modeling. We verify our
approach on various examples, ranging from an inhomogeneous anisotropic
stochastic differential equation on a torus, the chaotic Lorenz
three-dimensional model, and the Ni\~{n}o-3.4 data set which is used as a proxy
of the El-Ni\~{n}o Southern Oscillation.
