  We present a real-time second-order Green's function (GF) method for
computing excited states in molecules and nanostructures, with a computational
scaling of $O(N_{\rm e}^3$), where $N_{\rm e}$ is the number of electrons. The
cubic scaling is achieved by adopting the stochastic resolution of the identity
to decouple the 4-index electron repulsion integrals (ERI). To improve the
time-propagation and the spectral resolution, we adopt the dynamic mode
decomposition (DMD) technique and assess the accuracy and efficiency of the
combined approach for a chain of hydrogen dimer molecules of different lengths.
We find that the stochastic implementation accurately reproduces the
deterministic results for the electronic dynamics and excitation energies.
Furthermore, we provide a detailed analysis of the statistical errors, bias,
and long-time extrapolation. Overall, the approach offers an efficient route to
investigate excited states in extended systems with open or closed boundary
conditions.
