  In this paper, a data-driven nonparametric approach is presented for
forecasting the probability density evolution of stochastic dynamical systems.
The method is based on stochastic Koopman operator and extended dynamic mode
decomposition (EDMD). To approximate the finite-dimensional eigendecomposition
of the stochastic Koopman operator, EDMD is applied to the training data set
sampled from the stationary distribution of the underlying stochastic dynamical
system. The family of the Koopman operators form a semigroup, which is
generated by the infinitesimal generator of the stochastic dynamical system. A
significant connection between the generator and Fokker-Planck operator
provides a way to construct an orthonormal basis of a weighted Hilbert space. A
spectral decomposition of the probability density function is accomplished in
this weighted space. This approach is a data-driven method and used to predict
the probability density evolution and real-time moment estimation. In the limit
of the large number of snapshots and observables, the data-driven probability
density approximation converges to the Galerkin projection of the semigroup
solution of Fokker-Planck equation on a basis adapted to an invariant measure.
The proposed method shares the similar idea to diffusion forecast, but renders
more accurate probability density than the diffusion forecast does. A few
numerical examples are presented to illustrate the performance of the
data-driven probability density forecast.
