I will discuss a nonparametric modeling approach for forecasting stochastic dynamical systems on low-dimensional manifolds. In the limit of large data, this approach converges to a Galerkin projection of the semigroup solution of the backward Kolmogorov equation of the underlying dynamics on a basis adapted to the invariant measure. This approach allows one to evolve the probability distribution of non-trivial dynamical systems with equation-free modeling. I will also discuss nonparametric filtering methods, leveraging the diffusion forecast in Bayesian framework to initialize the forecasting distribution given noisy observations.